Can I replace my internal drive with a 320+ GB memory stick instead (MacBook Pro 13" c. 2009)
Yes, you can do this, provided your system will successfully boot from an external volume still. Failing logic boards don't always do this. Each time you boot you will need to hold the Option key down.

The speed will be noticeably slower, maybe even painfully so. The internal drive connects at 3000Mb/s. A USB 2.0 stick will connect at a peak speed of 480Mb/s with regular throughput being much slower. 

The largest USB stick you're likely to find is 32GB or 64GB. A 320GB+ drive will be a full portable external hard drive, much bigger than a simple USB stick. And bear in mind above all else that this is not a long-term solution. Your logic board will only get worse over time, not better.