how should I approach this networking opportunity?

Hello, to briefly describe my situation, I have an alumni I met at school alumni event.

The firm he works at is a boutique firm doing everything from IB to VC. He is a VP at one of the departments there and I visited the website and they don't seem to take interns on open, public basis.

The recruiting process isn't going too great for me right now , so I think my best opportunity would be to work there over summer. Otherwise, I might have finance internship in one of Chinese cities or do some marketing related internship in the U.S. or keep applying to boutiques.

While I think I kind of got along well with him when we had lunch at the event, it is not like I am close to him. I wasn't being very good with keeping contact. I still took it as hit or miss and offered to buy him coffee or lunch to talk about various things, and he said ok.

I guess if I were in less desperate situation, I wouldn't really openly express my interest in having internship there, but given the situation, should I be more open about it? What I am thinking of now is asking if I could work at his firm over spring unpaid, even just doing some administrative assistant stuff. He seems like a nice guy and I don't want to come across as a last minute guy who suddenly makes contact to get some internship (which I am lol) but seems like the only choice.

Or given that I am junior, should I try to maintain good relationship and have a shot when I am graduating (that is, if he doesn't mention it)? What do you think?