Starter overheating questions? 85 pontiac Fiero

Delivery driver here, just bought myself a used 1985 Fiero, and the thing runs great, But after a few hours of driving around town, starting the car several times and driving a few dozen miles around town.... the car won't start at all... until you give it an hour or so to cool down.

I assume that its the factory starter... i'm just wondering if you guys think this problem will be eliminated by a replacement starter or if this just isn't the right vehicle for the job.