I've converted bikes in the other direction. You'll need bars, shifters, and brake levers (possibly integrated levers / shifter), at the very least, plus bar tape. I suspect that in most cases you will also need all new cables.

It's easy to estimate the cost looking at niagracycle or Amazon, even if you plan on buying everything at your LBS.

The upgrade / replace question is always a dilemma. The economics depend a lot on your mechanical skills. I've tend towards making incremental upgrades, for years, but am actually thinking about getting a new bike at this point.