I got accepted to a Ph.D. program in New Zealand. For now, I have enough funding to cover only my first year of study, which is why I plan to work in that country while I study. Is it advisable to look and apply for a job there now? Or should I look for a job when I get there?
Two important points to consider:
First, a PhD is a full-time job. Most PhD positions come with a side-job that provides funding and, importantly, some sort of tuition forgiveness. Sometimes that side-job is fully overlapping with your research work, sometimes it's a separate task (like teaching), but in either case it's a position designed for someone doing work towards a PhD, with full expectance that their time will be split between those tasks.
I would highly recommend against taking any PhD offer that requires you to pay tuition and expenses. Universities may make these offers, but really they should only ever be taken by people who are already so wealthy that the cost doesn't matter. Consider them like a bribe or donation to let a rich family send their kid to a prestigious place. You really don't want to be in the financial position of paying a bribe meant for oligarchs and kleptocrats.
Most other jobs, besides hourly work, will expect your full attention. You should not expect to be able to hold a job and make satisfactory progress towards your PhD at the same time. Rarely, some people may work on "part-time" PhDs, but they should expect to take much longer to graduate and need to arrange this with their institution as a specific plan to move slower than normal, not just show up to do half the effort.
Second, if you're studying in a foreign country, you'll need to comply with their immigration policies. I do not know any specifics about New Zealand, but you should certainly not assume that whatever immigration policy allows you to be there as a student allows you to hold a job. It seems most likely you will (see https://www.immigration.govt.nz/new-zealand-visas/options/study/working-during-after-your-study/working-on-a-student-visa per comment by Allure), but you'll have to ensure that applies to you and doesn't conflict with any rules for being a student at your institution.
First thing you should do is discuss with your supervisor, because you can't really do anything without his or her approval anyway. Getting a job will not be easy - it's not that you don't have permission from the New Zealand government to work, you do, but rather that it is very time-consuming and that will detract from your PhD (which is also a full-time job). There's a good chance you'll get some supplementary income as a teaching assistant at your university, but it's not likely to be sufficient to live on.
So ultimately you will need to find funding somewhere, and whatever you decide on you will need your supervisor's agreement. Therefore discussing with them is the first step. You can decide afterwards if you want to, e.g., convert to a part-time PhD and spend the rest of your time working.