Fixing the lemon I just brought! Water pump question?

I brought a 2000 Cadillac deville 2 days ago. sadly I had a mechanic do a paid pre purchase inspection and the car is now overheating. i got it for a really good price so depending on the price of the repair it may be ok. the car overheats slow. I haven't drove it more than 15 mins at a time and it hasn't got as high as 3/4 on the temp gauge. but basically it takes about 15 mins of driving to start to overheat. it will hold steady at half sometimes. I took it to a mechanic there was coolant formed under the water pump and dripping. when the mechanic was checking the car prior to me buying it the lady was insistent on saying her boyfriend had just changed the water pump even tho the mechanic didn't even check or mention the water pump. i suspect she was well aware of the problem and must've received a high quote and put the car up for sale. anyways from the symptoms i described does this sound like the water pump. The radiator is overfilled with mostly water which again lets me know that the sellers were well aware of the problem. any advice? I asked the mechanic that said it was the water pump was he sure and would that fix the problem he said he wasnt sure. he was acting dumber than usual. he quoted me $600 I would like to know if my problem would be fixed if he's telling me to spend $600 i'm not rich! I must say the car drives really good other that that and I've been checking for drips and never saw any until the mechanic raised it up