2 grand is more than enough to cover commuting for a DECADE. You could get away with 1 grand if you wanted to. Cars aren't getting any cheaper so I only expect the bike to become more and more dominant.
Every bicycle I've ever owned has been bought from craigslist and some steel thing built in japan in the 1980s. Most bought between $400-$700. Many of them were quite nice too as Japan was the top builder of this technology at the time. They'll last practically forever.
The great thing about old bikes too is that if they're not too old (ie. 1980s and newer) they all use ISO standardized parts and are remarkably easy to repair and update.
Cycling is absolutely the most cost effective way to transport yourself around a city and incredible how the media so often frames the bicycle as some plaything for rich yuppies instead of the reality of being a critical tool of affordable transportation for regular low income workers.