The 20 ga is fine for the conditions you have mentioned. I would ask why the H&R Ultra and why 20ga though? They are excellent guns no question. However for the situation you mentioned it would not be my first choice,(unless the price is the major concern).
This guns accuracy comes with two drawbacks. First obviously is the fact that it is a single shot. That means your unlikely to get a second shot in the close cover you mentioned. No big deal if the first shot was where you intended of course.
Second and more important is the weight. I assume you have shouldered one of these guns. If you have not, you should. In my opinion these guns are unnecessarily heavy for a close range woods gun. If you primarily stand hunt, then it is Ok, but i would not want carrry this gun when stalking for any amount of time.
Of course none of this matters if you must have a 20ga. As the Ultra is really the only choice there. But why not consider staying with a 12ga. That gives you much better options as far accuracy vs utility.
I don't want to get off topic here....so If you would consider looking at 12ga's and want more info let me know and I will follow up with my thoughts there.