As a Research Associate I wrote a paper which was published with my professor line manager as second author (even though he didn't work on the paper). I did the research, I wrote the paper, yet his name appears first and mine second in all online repositories such as Google scholar etc. Is this right? What should I do if it is not? Should I be even bothered about this? In the actual paper PDF which can be downloaded my name appears first. It was a good paper as it won an award. But in online repositories my name appears second.
If you're the legitimate first author of the paper, I wouldn't even bother to find out why the error happened in the first place. Instead, my answer focuses on your question of what to do now.
There are two good reasons why you should insist on a correct authorship record: (i) First authorship of a (good) paper is a valuable asset. (ii) So are citations of your paper. With inconsistent information in the repositories and the paper itself, citation counting tools like that of Google Scholar might not index all of its citations. Keeping that in mind,
the most important party to contact is the publisher of the paper, since these people maintain the official record. In particular, they distribute the meta-data of their publications to other repositories. I can't tell if they will actually distribute individual error-fixes as well, as would be the most convenient option in your case. Still, they can correct the faulty data in their own repository, which will strengthen your case when you contact other parties.
Some databases/repositories, like DBLP, allow you to submit error reports individually. You can google for "(name_of_database) error report" or just look on the respective database's website to find details.
With Google Scholar, the situation is a bit more tedious, since they do not provide a process to submit error reports. Information usually gets checked and updated automatically after some months; there is nothing you can do to trigger this process. The thing you can do is to create a new, corrected entry for the paper in your profile. Then you can "merge" it with the existing entry, so that citations of the paper with its current authorship information are counted.