Alright, I made an account JUST for this thread! I, too, am having a bit of trouble with R.O.B., when playing as the Ice Climbers, but I've managed to figure a few things out. I'll help you out as best as I can, and I'm praying that others can lend some tips, too. But before all that, I'm just going to say this, now. I'm very, very far from being a pro-level player. These are just some things that have helped me out.
Whenever ROB is above me, I ONLY use the Up+B attack (sorry, I'm not extremely familiar with the move names and terms). This sends Nana into ROB, preventing you from taking a hit of his Down-A air attack, while racking up a little bit of damage. The only downside of this is, of course, that you're unable to perform any other action, and has a bit of recovery time.
When ROB sends out his Spinning Top attack, you can stop it in it's tracks, easily, with the Blizzard attack. I usually do this if I see him charging it straight ahead of me. I believe it will also allow you to walk over and pick it up (just don't get too close!) to throw back at him. I think the Ice Blocks also work for this, but I swear by the Blizzard attack.
If possible, try to avoid maps such as Luigi's Mansion (Rob's Up-Smash attack is awful!) or maps with similar low-leveled platforms. Trust me, it can be awful.
Another tip is that his Up+B special needs to be charged, like two of his other B-moves. It charges by him being on the ground. So, if possible, try to keep him up in the air. I know this can be one of your main focuses, anyhow, but I figure it's worth mentioning.
The last thing I can think of is that his Up-Smash attack will usually get you, if you're above him. If you're falling, and he's below you, I wouldn't suggest trying to do a Down-A. Not for just the fact that his attack has higher priority, and will beat out yours, but also that he can dodge yours easily with a side-roll, and hit you with an even more powerful attack, while you recover. The only thing I can really think of doing, off the top of my head, may be just trying to get him out from below you. Maybe just try to avoid him, altogether, like with a Right+B attack.