My door doesn't close without having to push it upwards first
The door is sagging, as is common over time. Or the house is sagging, and the frame's gone out of square.

In either case you have a number of options depending on what's wrong. Closing the door you should be able to see where the gaps are, and likewise with the door nearly ajar.

Then shim the appropriate hinges. In a perfect world you'd shim behind the stud, but that requires opening the wall and I'm guessing you don't own the house (and it's a pain even if you do). I've used a few layers of cardboard as a thin hinge before, or you can use real wood shims if you have a huge gap to make up (use them in pairs, and snap off the parts you don't need). Obviously the screws have to have enough to bite into even with the shims in place, or you might need slightly longer screws.

It's also possible the wood that the screws are going into is weak and can't hold a screw; you may be able to recover this by shoving toothpicks and superglue in the holes, then redrilling new holes once that dries.

If the house is sagging, you may be able to straighten the frame through brute force. I didn't think this would work, but it made a difference in the house I used to live in. It was a non-load-bearing wall and I literally just pushed the door frame back into square, shoving from corner to corner on the shorter diagonal.