Driving in 3rd gear instead of fourth will not affect your transmission as much as your engine. In fact, if you are driving in a manner that causes a lot of shifting between 3rd and 4th, it may be easier on the transmission to leave it in 3rd. If your transmission needs fixed, I don't think it would be from leaving it in 3rd gear. As far as cost, it will vary greatly depending on what needs done and what type of car you have.
Assuming it is an automatic transmission, I would check the fluid level. When the transmission is warmed up, the level should read 'full' on the transmission dipstick. Check the color and smell of the oil. Does it smell burnt, or does it appear to be brown instead of red? Perhaps the filter screen is clogged.
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