Find a cheap hotel as central as possible so you can walk around manhattan and get back late at night without walking through quieter, isolated neighbourhoods.
I love NY for walkability. It’s to best way to experience it in spring!
Of course you’ll need the subway / ferries to see some key sights. The Staten Island ferry for great views.
Cheap food can be had by eating at diners, bagels, street food takeaways (pizza, Chinese, Thai - the street hotdogs aren’t good though!), getting a supermarket picnic for your hotel room. Some bars give free hot food snacks in the evening with a drink.
The bars in Grand Central Station are iconic and good value.
I recommend the free museums and galleries in the day as well as exploring Central Park. The Metropolitan Museum on the east side of the park. The museum of natural history on the other side. There are many more! Museum cafes are usually good value for meals, and often beautiful with nice views.
Mostly just walk around the city. The High Line is gorgeous. Free and the best way to soak it all up. Wear very comfortable shoes!