Disclaimer: I ride but have yet to personally encounter a practical situation to ride to a fishing locale.

My first inclination would be to mount a rear rack on the bike and use the rack's frame to secure the rod tube vertically. The rack would also hold a pannier or top mounted bag where I could stow the rest of my gear for the ride.

Riding on a mountain road, I want to be free of unnecessary distractions. I definitely wouldn't mount a rod across the handlebars, and I wouldn't want one knocking about on the top tube. In the absence of a backpack of sorts and a shorter rod, I'd prefer the rod mounted vertically and behind the rear axle and drivetrain.