I have found grip shifts to have more problems over time than trigger shifters. If you plan on riding this particular bike for several years, I recommend swapping out the revoshift in favor of comparably priced shimano trigger shifters. Since you are even considering taking apart the shifter, I believe you will be able to install new shifters, cables and housing. Remember that new cables will stretch quickly, so if you choose to change shifters, be prepared to adjust the cable lengths after the first couple rides.
If you would like to fix the shifter you have, make sure to take a picture of the shifter mechanism when you open it up before starting to work on it. Then make an outline of what you see on a piece of paper and tape to it the screws you remove in their respective locations. Make new such outlines for each level of shifter as you remove more pieces. This way you can avoid losing tiny screws and springs and other pieces and you will know what goes where when you try to reassemble the shifter.