You can ask them to throw in a small ice pack (called horeizai in Japanese), although I will tell you that many ekiben stands on the train station concourse or platform won't have them. If you go to the basement deli sections of major department stores and buy bentos there, they will have horeizai's. But such places won't be open in early morning.

Anyways, these ekibens seem to be good for a while at room temp. Of course, try to keep it away from sun and heat. I recently bought a bento with real crabmeat on top of rice at Sapporo Sta. I asked for horeizai, and of course they didn't have it. But the lady pointed out the good-thru date/time at the bottom of the bento, which was something like 10pm that day. She said there's nothing to worry about as long as I kept it in room temp and ate it by that time. If you're really worried, perhaps you may be able to snag a bento at Shin Hakodate. I've never been to Shin Hakodate station, but I presume it's not huge and most major train stations have some place to be able to snag a bento during a train transfer.