Veteran renter here - here's my somewhat random advice:

• A month is enough time to find a place, as long as you're not extremely picky. But start looking the moment you are able. It is a competitive market.

• There will be a lot of people looking for June 1, so if you see a place you like, snag it quickly, or else someone else will.

• How much you can afford for rent depends on how much you'll spend in other ways. But on 120k, as long as you're not hugely financially irresponsible, you should be fine. Stay under 3k/month, at least. The further under, the better.

• Not every place requires four months rent up front. Some leave off the last month's rent, or the security deposit. But you should plan to have at least three months ready if you're going to be working with a realtor (they take a month's worth for themselves, if the landlord doesn't cover it, which generally they don't).

Best of luck - hope it all works out for you.