How much is the cost to fix a rump-sprung chair?

[Paper Roses]

3. Thru the years we had many chairs and couches retied.

If you can up-end the chair and the base of the seat is covered with(Cambric) fabric, use a tack remover and pull enough of the old nails so you can take a peek. If the springs are visible, a good upholsterer can retie them from the bottom up. It is not always necessary to tear apart the whole seat if the springs can be reached. Bring it to your favorite upholstery shop after you peek and see what can be accessed. The cord that ties the springs might be broken from age. Don't know what you might see in regard to any padding but it certainly is worth a look-see by someone who knows.

I'd ask if they would take a look at see if it can be done bottom-up.

You can staple the old fabric back if you decide not to proceed. Careful, old Cambric is fragile. Check this pic.

Good luck.
Edited to add, the webbing may be torn and that can be replaced via the underside too.