The best thing to do would be to drive it (stay close to home) and try to duplicate the problem the previous owner described. When the car stalls out, take it to the nearest autozone and ask them to scan it for trouble codes. They don't charge for this, and they will know exactly what you mean. Come back and tell us what the trouble codes are and we can get you on the right track.
What you describe sounds like a bad fuel pump to me, but I certainly would not want you to install a new fuel pump based on the limited information we have. As John mentioned, another possibility is a bad fuel pressure regulator. Here is a picture of what your FPR looks like:
At one end of the FRP you can see the gold colored fuel line. At the other end is a black plastic piece sticking out that is about the diameter of a pencil. On your car there will be a rubber hosed hooked up to that black plastic piece. Pull that hose off (it may take a little persuading) and smell the open end of the hose. If you smell gasoline the FPR is defective and should be replaced.
Here is a YouTube clip that shows you where to find the FPR on your car:
Check the FPR, and get the car scanned for trouble codes.
2001 Bonneville SSEi (retired at 365,000 Kms.)
2014 Jeep Grand Cherokee Overland (yes, I know its not a GM)