I wouldn't put money towards that bike, yes you probably could swap out the shifters and adjust your derailleur but in the end I wouldn't put much into that bike. Also this is not really a mountain bike question it would make most sense in Bicycle Mechanics subform,

The bike is probably worth the price of the new shifter, and new cables and housing as well as labor if not a little less depending on the condition of the rest of the bike. You are better served saving that money towards a nicer bike unless there is something broken on this bike and you refuse to get a new one which isn't a great idea.