If people do need something, they move heaven and earth for it.
So people, somehow, using horses for transportation is fine.
However the real question is does it make sense?
A post apocalyptic setting would mean a couple of things.
For starter cities are a lot smaller with a situation similar to medieval, particularly medieval, cities.
As people can't control, defend, and supply the mega cities of today.
So the scale of a city is never going to be an hour drive to get to work.
You just can't maintain that sort of city anymore.
So it changes into 20 minute walk to walk.
Also cycles are a much much better alternative to both cars and horses.
Simple, easy, cheep, portable...etc.
And when the whole new city can be covered in a 30 minute trip, people would rather use them.
Second important point is that horses, historically or not, tend to be expensive.
You need to constantly care for them and make sure they are fine.
For example. You know parking lot? Well. If, in that world, you park your horse in a place, it needs to be cared for.
Stuff like the heat, wind, snow...etc are bad for it.
Then it needs to be feed and watered the entire time you are doing your work.
At home you need to have a stable or suitable place for it to stay.
And in PA setting where land is already limited you can see how that would be a problem.
So in essence. The logistics of using horses inside cities in PA setting don't make sense and bicycles are much better.
Now if you are thinking about travel it might make more sense depending on the setting. However days of horseback riding to reach the nearest settlement is not fun. It's long and exhausting.
Then if you factor dangers on the road it becomes much worse.
Cars would still be used until there is no more fuel.
But the thing is that in a PA setting people could just start conserving fuel to for the out of city uses and have enough for trips.
Then we already have electric cars which can still be used.
Then you need to start factoring dangers on the road.
If you have to deal with mutated creatures then can horses be safely used as transportation? Do you need to deal with people with guns?...etc.
So I guess my point is that it's not a matter of can people do it or not.
It's a question of would people do it or not.