Columbus has zero rail lines and busing is anemic.
It is a rust-belt metropolis with some world-class research facilities, (OSU, Ohio Supercomputing Center, Battelle, Chemical Abstracts), a lot of banking (Bank One had its headquarters there before Chicago First bought them out, and there is still a huge IT presence). Columbus is often a test market to try new product lines -- it is large enough and _boring_ enough.
Amazon coming in with an HQ would inflect the whole metropolitan area. Everything in Columbus right now pivots around the Ohio State University and the network of research centers, but that center of gravity will probably shift to Amazon if they come in. That is, Amazon would have a huge say in how they want the city to be developed.
As for rail: One thing from Atlanta's playbook: create greenspaces out of old rail right-of-way. Some of those may be converted into light rail. Columbus's High St. (the central corridor running down the spine of the city) used to host a trolley line. It's been the target of gentrification, from the Downtown core, through the Short North, and into the OSU campus area. Restoring a light rail line there is not out of the picture, particularly if it is extended to the airport.
I only know all of this because I grew up there, and my 6th grade social studies course was on Ohio history. That is local knowledge.
I don't know whether Austin can do anything, but I'm pretty sure there are some unique local features that can be just as much of an advantage, things that the locals know but is not so apparent for anyone else.
I still think it has a lot more to do with how willing the city is to have their development plan influenced by Amazon more than anything.