Hi Zeldas, I'm from the Robin board, and saw you were discussing this MU.
First off, about customs - I'd probably just run default Robin over customs vs a Zelda, though his customs give different options in other matchups. I'm not familiar with Zelda's customs, so everything below this is going to be about them both without customs.
Robin pins down Zelda's neutral quite well. Zelda might have a reflector, but this doesn't really pose a problem to an aware Robin. Arcfire being reflected is no big deal, it'll just bounce up past Robin, neither hitting him nor giving Zelda time to attack Robin out of it. I fought a Zelda in tournament who would actually DI out of Arcfire's initial hits, and then reflect the last hit, to try and prevent me from following up after hit confirmations. However while this surprised me the first time, it's a wildly unsafe option - the next time I ran up, shielded the reflect, and fsmashed him out of it. Arcfire's main weakness is that a lot of characters can just roll around it, leaving Robin open - yet Zelda has a poor roll, making this not as viable for her. She also lacks the speed to run past him.
In terms of reflecting thunder, all Robin needs to do is be in a position where he could toss it out, or he could just play the normal neutral game, where throwing out Nayru is unsafe unless it's a very hard read. Basically: unless a Robin is trying to camp you super far away (he shouldn't), reflector isn't going to help in this match.
Robin does fall into some combo weight for Zelda, such as her dthrow to Nair, dtilts and utilts. Dash attack is also a good option against arcfire, because Zelda ducks which can avoid its initial hitbox. Robin's linear recovery can make it more difficult for him to get back to the stage.
I had two sets recorded against Cosmo's Zelda:
https://www.youtube.com/watch?v=ghMQQEcyZQs
https://www.youtube.com/watch?v=TAcU9iPJG68
I actually played another Zelda earlier in the pools bracket too, unfortunately those weren't recorded. This is probably not going to be a commonly seen matchup in tournaments though.
Just due to Robin's options in their neutral game, I'd put this close to even but slightly in Robin's favor.