GGs, Zig.
Here're the major things I noticed:
1. You were, like the others said, a bit aggressive. You didn't give yourself time to reposition after getting hit much of the time. That is, if you got hit, your immediate response was frequently to hit back. (though you did a good job)
2. You could have punished (or at least made him more cautious to use aerials so close) with OoS NAirs.
3. He approached almost exclusively with NAir. If a ROB hits your shield with NAir, they'll usually either roll or jab -- he rolled. If they hit you with it, they'll follow up. This was what I saw you get hit with the most.
Something I find really effective versus ROB is Aerudo. ROB's mobility isn't so great, and being a big target, if you have him on a platform (you did do an Aerudo against him on a platform which was great) or near the ledge, Aerudos are amazing to get by gyros and surprise them. Beyond that, I think against characters like ROB, the biggest thing is to find out out they react to things: jump-ins, airdodges, etc. Jump-in mixups are good vs ROB, I feel, and throwing SH BAirs to scare them is good, too. I wouldn't use BAir with the intent to go out of my way to hit, rather, just to send the message that he runs the risk of running into the Fist of Evil.
My MO in this MU is to get to about mid-range against ROB. This is where he's forced to either move away or start attacking. Then I'll either do jump-in mix-ups, or challenge his aggression.