Nope, driving is must if you want to do the horse caring. In the past you had to know how to ride a horse (these days, funny enough you do not need to know it - at least not the extent to be good at it). But you need a transportation there.
And horses just like other big animals tend to be kept in quiet areas, where they have less traffic stress to begin with. At least that is the setup in most countries with a waste land mass.
You plan to learn to drive there is not a very good one. Unless you have an experienced driver to be around you all the time, you will have a hard time to proceed. For a beginner driver (G1) it takes at least 8 months (that is after passing certified driving sessions) to be allowed to do the driving test.
In reality count for it 1 year +.
So by the time you can do it, you will have to move to the second town.
One way around it would be to go for a motorbike. In that case you can get the needed driving licence much faster.
As for job opportunities, there are no many there. (it is a really small town). So you will have much harder time to get something. Maybe save for the work on farms (that might be more like a seasonal job).