I would like to submit my manuscript to a journal. Before I do, I would like to reach out to a professor at another university to see if he would be willing to review my manuscript since he is an expert in the field.
Is this appropriate or common practice?
Ask your advisor at your own university, but the answer is 99.9% no. It's your advisor's job to look at your paper before you submit.
In some cases it's appropriate to share with outsiders when they aren't really outsiders but rather collaborators or colleagues known to you or your advisor. In other rare cases you might reach out to someone whose work is very similar or affected by yours, but again, this would be at your advisor's direction.
edits:
OP clarified in a comment that their advisor has reviewed their paper and found it suitable, but the paper is outside the advisor's best expertise. In this context, I do think it is worth getting an outside opinion if the advisor agrees, but I still would not suggest cold-contacting anyone.
Instead, I would recommend asking the advisor if someone in their network might have more expertise in the specific area, even if they aren't the very top person in the field. The point is to get an informed opinion from someone friendly but willing to be honest.
Another option is to do some networking at conferences, etc. In that way, you turn people you've never met into people you've met and it may become more appropriate to ask (and certainly more likely to get a positive response).
I am surprised at all the "no" answers/comments here. This may depend on your field, but in my area (pure mathematics), it is very common to send drafts of papers to experts for feedback/comments regardless of whether or not you know them. (I presume this is what you mean by review, rather than anything more formal.) Now posting papers to the arXiv fills this role of soliciting feedback to some extent, but it is still common to send your paper (often with a link, but sometimes a pdf) to individual people for comments.
That said, often you will not get feedback, but in my experience your chances are much better than 0%. This may be because the other researcher is too busy to look at your paper, or because they don't have any noteworthy feedback. One thing to increase your chance of feedback is to ask a specific question about a point in your paper you think that person may have insight on.
Also, if you do not know the professor, you can preface your email with a sentence introducing yourself (e.g., I am a PhD student of Professor X at StackExchange University), which may increase your chances of getting a reply, particularly if that professor is close to Professor X. If you do have an advisor/mentor, you can also ask them advice for who would be good experts to ask for feedback.
Academics are very busy people, so it is highly unlikely that this professor will agree to a review outside of the journal process. Ordinarily, a preliminary review would be done by your supervisor or a colleague that does not mind taking the time to help you, and after that you would submit to the formal review process for the journal you submit to. Bear in mind that you can often suggest reviewers to the journal during the submission process (either in the online form, or just in the body of your letter to the editor), so if you think this professor would be a good reviewer, that may be an avenue to get a review from him.
Making a polite and humble request, with proper introduction of yourself and giving the possibility to say a gentle "no", is by all means appropriate. I have a been in this situation, and my experience says, 99.9 % of the time its a no reply and occasionally you may get a "interesting but not close enough to my area", or "Sorry I am occupied with too many other things now". You can mail it anyway, nothing wrong. (Also add to arxiv.org and share a link to professors rather than sending a pdf).
My suggestion is, review it yourself many times, put yourself in the place of a reviewer and make judgements and estimate short falls. Iteratively improve presentation and polish it many times. (this could include adding redundancy to give more clarity on central ideas or contributions, including references(that you were not aware of when you first wrote the manuscript) to earlier works and giving comparaitve statements (these are the ones reviewers request almost all the time), extending your work, or writing about its applications that you come across while reviewing, and so on...).
When you have done these things repeatedly over a few spells, start looking at journal to which your paper matches the best (in terms of scope and subject matter and nothing else) and submit. You will get a thorough an professional feedback. Use it to improve and submit it to else where and repeat till its published. Meanwhile you can add it to arxiv.org to get visibility.
PS : Do not submit to very high on prestige journals. Also do not take turn around time into consideration while selecting journals. Select solely based on scope and subject matter.
I'm also surprised at the feedback. I think what the original poster is getting at here, is that having several "well-known experts in the same field" perform a review of the paper is a requirement for submission to the publication. This is something my wife has run up against on papers she has submitted, and while I don't know about "Academics", researchers and well known scientists are familiar with this process. Generally, they will ask you to perform the same service for them in the future on their potential submissions to the same or other journals.
That's not to say that they "must" agree, or perform the review when you ask, but you should definitely come up with a variety of potential reviewers, and use the law of averages to your advantage. The earlier advice of having your supervisor or the principal investigator solicit reviews on your behalf is also good advice. This will become less of an issue as you become established and recognized in the field, but initially it's a challenge. Make sure to give a good overview of your research, the people involved, and the institution(s) sponsoring the research, so that whomever you contact understands this is not some fly-by-night effort for a complete stranger to get recognition they don't deserve...After all, in essence, you're asking a complete stranger to vouch for your work using their good name and credentials.
Peer reviews are important, but as a new entrant into a field, it can be very challenging to get the ball rolling.
Is this appropriate or common practice?
Appropriate, yes; common - I don't know, but I've done it. And with a monograph even, not just a journal paper...
I'd make it at least a two-phase process, i.e. don't start out by making the full request and linking to the paper, but rather make it a more fuzzy request; also, try to make it seem like it should be interesting for him as well, e.g. by relating your work to his.
Of course - it is always better if you can approach him in person at a conference, or if you have a common acquaintance you could ask for advice on this.
PS - It being appropriate does not mean that he will say yes :-(
