The trolley is good all the way out to El Cajon (have taken it from La Mesa to Petco Park to watch the Padres). I think @knoxblox will have more specific advice using the north/south line (TJ to La Jolla), but it does go right next to Old Town. I haven’t used the buses in SD, but now I’m curious and want to try using them to get to Balboa Park.
I haven’t used the trolley system other than up and down the main downtown corridor. Sorry.
I noticed that the Kona Kai Resort at the end of Shelter Island is running specials this winter. It’s away from downtown quite a ways but it pretty affordable for what it is at a bit over $200/night. We’ve stayed there before and it’s really nice!
I can’t speak to the bus system. Though a 4th generation native, I haven’t lived in San Diego for over 30 years, and have only sporadically visited over the past 20. The trolley was quite pleasant and usable when I was there. If you’re really staying as close to downtown as you describe, I would hazard a wager that you’ll probably be fine with trolley, buses, and occasional Lyfts/Ubers.
Most Januaries there are shockingly pleasant, though being right next to the ocean you might get some morning fog. That said, if this ends up being a wet winter, January is when that will happen. But I would bet against anything troublesome. Last time SD had truly nasty rain in my memory was 1980.