Airbnb is a well-known global web site, and it has testimonials so you can search for what other guests mentioned about the apartments and their owners. Note that some are b&b-type home stays rather than apartments with sole occupancy.
I keyed in your rough dates and found a one-bedroom apartment in East Village for 101 an evening; residence stays are usually cheaper.
Choosing good short term leases in NYC is going to help you make sure that you are going to have a cheap place to stay that is not going to get you into a really long lease. You can go for a couple months on these places, and you will be able to get the exact things you need from these places when you are searching in the right spots.
You should check short term rental in NYC for short term options, and you will find a lot of people who are subletting their places for a short time or will let you take a short term lease. This is so much easier for you to handle than a normal lease, and it is going to save you a lot of money over the life of your lease. You should not have to spend too much, and now you can pick something short term that works.