Zaldog-
The cutlass bearing supports the propeller shaft as it exits the boat. It is usually bronze with a rubber sleeve. Through use or engine misalignment, this rubber wears out, and the shaft can wobble as it rotates.
How much it costs depends on the type of boat and the boatyard. It will vary, depending on who is doing the work, time of year, etc.
It isn't all that hard to do if the boat is out of the water and you have a modicum of capabilities with hand tools, and are willing to sweat and bleed for your boat. Here's how to do it roughly... it may vary a bit depending on your boat, but this is the basic idea.
Remove the propeller and shaft. Now, with hacksaw blades, cut through the rubber and bronze of the old bearing. You can't use a proper hacksaw, as the blade has to extend into the middle of the bearing. Make a second cut close to the first, and with pliers (or whatever) remove this little wedge of bearing. What you are trying to do is take a small section of the bearing out so that you can decrease its diameter. This way, you can eventually pry and wriggle the old bearing out.
The bearing is supposed to fit very tightly in the casing. You will have to use a hammer and a block of wood to get it in place.
One trick that can make this a bit easier is to freeze the bearing first. This will make it shrink just a little bit and it may go in easier. With a wood block over the end, drive your new bearing in with vigor.
Remember, friction causes heat, so do this part quickly. You may not get it driven in all the way. If it's less than an inch, just cut the exposed part off with a hacksaw. Now put the shaft and propeller back in.
Be aware that a worn cutlass bearing could indicate improper engine alignment, so re-check this when the work is done.
It is quite possible that your shaft has worn too. Check the propshaft, and if it is scored, then get a new one or you'll end up doing this all over again at the end of the season.