Yeah, it’s hard if the budget is literally zero, but if not, students get affordable monthly tickets to public transports at least around here, so that already makes it very possible to get around on the cheap. It’s not expensive to start with, at least anywhere where I’ve ever been to, as a local or as a tourist, so only in very extreme cases would that be a problem. Even then, asking around one can get lucky and get a used bike on the cheap, or for free from a relative or a friend. Depending on the city, the bike can be faster and more convenient, even, than public transport.
Once you can get around, a city turns into a treasure trove of possibilities and activities. Sure, there are expensive options, but also a huge variety of free or cheap options too! The toughest part is daring to go. It takes some bravery, especially if doing something or going somewhere for the first time, but it always gets easier and easier once you get through those first hurdles.
Edit: To add, there are almost always interesting places to spend a day in just outside the city proper. Or even a bit farther away, but still along the public transport routes or bike’s reach. Where I live, we have some public apple orchards for example, one can make a day out of going to pick a basketful of apples and having a picnic. In the winter there are fun slopes on some smaller hills that have publicly maintained slopes to go sledging in, make it a day with some hot chocolate and friends! Some smaller museums outside the city center are very affordable and can have quirky subjects and presentations. Even just biking or walking to some lake or park can be great, take some acoustic instruments or a boombox or something, sit around and chill, alone or with friends!
For things to do alone, at least in my city, we have public outdoor gyms spread around. Some of the farther ones don’t get much use, so I can usually be alone there when I go, it’s very zen especially in autumn when it’s a bit chilly. Local coffee shops and some bars have free exhibitions of local artists usually, same for libraries. Sometimes you can get lucky and have impromptu live music too, with no additional costs! Smaller local bands have very affordable tickets too, usually less than a pint. And if you’re a student, at least most places I’ve ever been to, you can get drinks on the cheap, if you’re not too picky about your vices.
This is all not even mentioning things like tabletop game groups that gather just for the fun of it, no cost. People bring games and you play, no need to own any yourself. Usually it’s coffee shops or bars, so there might be costs from the aux things like drinks, but you can participate with just water, who cares? Often these are advertised in said coffee shops with some printed paper on a wall, or maybe the various boards across the city and markets and whatnot that have adverts/notices etc.
The toughest thing, all in all, is daring to go. It really is the hardest part to be brave enough to go and do stuff. It’s not easy for anyone, but once you dare the one time, it gets easier from there, and gets fun very fast!