Q: Car is not starting up or staying running when jumped.

I got gas the other day and parked in my driveway, when I went to drive it after a few hours the dash lights were dimming and the fuel level decreased until it was empty and the car died however there was no smell of a gasoline leak. I got it towed to my house and a couple days later tried to jump the car, it started with a jump however peetered out after I removed the positive charge. I just had my alternator replaced with a refurbished one so think that may be the issue, however my battery is very old as well. My mechanic tried starting it yesterday and said it wouldn't start at all and that he thinks it's the starter but I'm not convinced and want to be sure before I proceed with buying parts etc.

My car has 212000 miles.
My car has an automatic transmission.