Tbh I don’t think you can find anything in the city, especially downtown near NYU, for under $100! Unless you share a room with someone or super luck out on Airbnb.
EDIT: jk actually you should look into hostels if you’re really strapped for cash! Idk if there are any near NYU but it’s so easy to get around Manhattan with subways or taxis so even if you find something on the west side or uptown it wouldn’t be too hard to get to NYU
Another edit: I’ve never stayed in hostels in the US tho, only abroad, so idk how gross they might be but I used Hostelworld a lot when I was abroad so try looking there!