Yes, use a press for removing and installing bearings. Figure out what replacements will work and sources for them before proceeding if possible.
I definitely understand not wanting to spend a lot on tools but there are definitely times you can spend more fixing something up than it would cost to buy a good one in the first place.
Lathe headstock bearings can be fiddly of preload needs to be set, also there are different grades of bearings available. Depending on the precision required it may be difficult to make a repair that will be satisfactory.
Would like to see a video and pictures if possible.
John