Techstars: One month left
postmaster.io
postmaster.io
I would say that learning to manage the mentor meetings helps you get way more out of it. If you don't do it well, you dwell on the same points every meeting without moving on to new stuff.
You're right that it's hard to build much during the first month. But when the first month is over, you feel very confident about what you need to do. That makes the next month and beyond incredibly productive on product and customer development.