I am intolerant of cold weather so I'd stick to Andalucia or the east coast. You may well be different in what you consider acceptable.
Albarracin is at an elevation of almost 3900' (about 1200m). I'd expect it to be rather cold in January. You can take a look at actual, historical, day-by-day weather statistics for Teruel on the website timeanddate.com. I've given you a link to the January 2021 data, but you should check out more than one year. (The website's data goes back to 2010.) I don't know about the prevalence of snow, but the area is rather dry; the climate-summary chart in the Wikipedia entry for the regional capital of Teruel shows not much precipitation at any time of year.
I can tell you a bit about the logistics involved in getting to Albarracin by public transportation: First you travel either to Zaragoza (which is on the fast AVE line between Barcelona and Madrid) or to Valencia (which has AVE service from Madrid and decent service from Barcelona). Then you take a regular train to the regional capital of Teruel. There are four trains a day from Zaragoza, taking between 2 hr. 55 min. and 3 hr. 30 min.; there are only two trains a day from Valencia, both taking about 2 hr. 25 min. Teruel is under-touristed and quite picturesque with its mudejar towers. Then you take a bus to Albarracin; there's just one per day, in the afternoon; I'm not sure whether they run seven days a week. ( I assume that bus is critical to some of the residents of Albarracin, but I don't know that it's currently running.) The return bus is in the morning, so spending one night means quite a short visit; even if you weren't hiking, you'd be rushed. There's an obvious risk here: What if the bus doesn't show up? On a really short trip like this, it would be a big deal.
There's a mostly-public-transportation option that skips the bus leg: Take the train to the small town of Cella, which is closer to Albarracin than Teruel is, and then get a taxi to cover the last 16 miles. I've never tried that; you'd need to pre-arrange the taxi. Cella has a population of just over 2500, so I don't know what the taxi situation is.
This is not an area I would recommend anyone try to visit by public transportation on a short trip even if I could guarantee everything would run on schedule. There's just way too much travel time involved.
However, weather issue aside, I think this area would appeal to you. I met an American who moved to Albarracin partly because of the rock-climbing opportunities, so that's another activity that can be pursued there.
With a car things would be easier, but you should check the driving time to access the area from Madrid or Valencia. A car doesn't move nearly as fast as an AVE train. But once in the area, you'd have access to at least three very interesting and quite different places: Albarracin, Teruel and Cuenca. Cuenca's on the AVE line running between Madrid and Valencia. It has a medieval historic district, a cathedral and a couple of small modern-art museums. There's at least one car-rental agency in Cuenca.
In the meteorologically more-desirable area of Andalucia, there are both hills (the Alpujarras, which have some bus service, at least in the summer) and mountains (the Sierra Nevada) near Granada. Granada itself is located at 2400', so it's not the warmest part of the region. With the time you have, you could have a nice visit to Cordoba (which would be warmer) and Granada, possibly with a side-trip or two. I mention those two instead of Seville because the latter is a much larger city. It's very picturesque, and the Alcazar is just one of its many great sights, but it does feel like a big city. I felt more of a small-city vibe in parts of Cordoba and Granada.