This is the right approach, and it could make your bike much more comfortable, assuming it's not way too long currently.

They come in a wide range of lengths, from about 60mm to at least 130mm. (MTB stems come even shorter, but the ones for road bikes start around 60, maybe a little shorter). You need to make sure to get the right size for your steering tube clamp (usually 1-1/8" for most quality road bikes made in the past decade or so) and your handlebar (again, for most modern bikes it's 31.8mm, but don't assume--measure it, or see if it's marked). They also come in various angles, and you can choose to angle them up or down according to the riding position you want.

All this assumes you don't have an older bike with a quill stem. Those are a whole different deal.