If it is in written English, it is fine for me. In case it is in spoken English, the sentence sounds kinda long for me. I would say "I'm a 50-year-old small business owner. I'm going back to college to finish up my undergraduate degree". Do you think native speakers would tend to say so?
Is it wrong to use " that" instead of "who"? I think both "that" and "who" are okay.
If someone has only a bachelors, they tend not to refer to it as their "undergraduate degree."
Only when someone has a graduate degree do they tend to refer to "their undergrad."
I got my MBA from Temple. I got my undergrad from Villanova. Or whatever. (Neither of those statements are true, by the way.)