I use neumático personally (Spain Spanish) and along with llanta those two are the most common words. According to the Wiki article on neumático, you'll also hear cubierta, goma, or caucho in other countries, but it does point out that llanta is the most common in Central America.
Taller mecánico to me is pretty much exclusively automotive-type repair (again, Spain Spanish), but apart from adding in farm equipment repair (or similar), I can't imagine anyone in the Spanish-speaking world not understanding what you mean with by that, especially if you've already given the context of a busted tire. Also, from looking on-line, it seems that taller automotriz is used and even often in conjunction with the former as taller de mecánica automatriz.
Reparar works just fine for tires and greatly prefered over arreglar and a quick search found not much evidence for using zurcir (someone will doubtlessly point out in the comments if they've heard it)
So some possible questions that'd be pretty universally understood (again someone will point out in comments if not) would be:
- ¿Dónde está (o queda) el taller mecánico más cercano? (following Emilio Gort's format)
- ¿Dónde se puede reparar una llanta pinchada?
- ¿Sabe(´s/s) dónde hay un taller mecánico? Mi llanta está pinchada.
Make sure to add in pinchada — that way even if llanta is used for something else like the rims, you'll give more than sufficient context to be understood.