The wheels are the most expensive part, if the wheels are good you can't go wrong.
If the shafts are broken or rotten, they are around $100 plus shipping.
Dash looks good.
I think this cart is a steal at $350, if the wheels are sound. They are very old with the staggered spokes.
Check for rotting in and around the hubs.
Grab the rim of the wheel and pull it side to side and see how much it wobbles back and forth on the axel, some is ok, but you don't want to much. you want to make sure the wheels go with the axel too.
Wobble the spokes back and forth at the hub and at the rim to make sure tthey are tight.
Check the rim of the wheels, is it tight, and check the rubber on the wheel for tightness.
Every thing else is an piece of cake.
Some wheel wear you can fix but rotten spokes and hubs are expensive.
Put the axel on a jack and spin the wheel, see how many times it goes around and that it doesn't wobble to bad, a little is okay.
What size horse do you have, I can't remember?
It looks like it is for a full size horse, but you will need to measure.
It also makes me wonder why no one has bought it yet.
If you do all the work yourself and there are no major problems, I would say you could get by with $350-500, paint, sandpaper, stripper, new patent leather for the shafts, pieces of wood here and there (spindles etc.), new bolts and nuts, new seat cushions and a new seat back, and lots of elbow grease.