See Contemplative Wanderer.
This definitely feels wrong in red, but if I'm making a cycle I suppose this makes the most sense as a phase red would skip. I suppose it would make the most sense as a drawback and not as something red would inflict on others.

I wouldn't say it feels wrong, per se, just that the rules text wording doesn't do it justice. If there was some way of saying, "enter combat immediately after your draw step, then end the turn" it might feel better.

But still, I really like it. This dude comes down and is like, "Fight! Fight me! Fight me now!"

All in all, this comes together in a nice package, wrapped in flavor and sealed in win. My only issue is that 'Rushing' could be some other term or synonym to make him seem more aggressive.