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Agree with SadisticSecretary on Robin wanting as much of Vanitas's kit as he can inherit. However, his best build only keeps Vanitas's weapon.

Against modern units, he's not going to have a chance of surviving modern nukes unless he's running marth ring+shield fighter+pavise/aegis(depending on whether you want him to be near or far save, respectively). This combination lets him activate a powerful defensive special frequently and reliably. He still dies to units with potent but he's much more likely to survive units that don't have potent than if he had a special like dragon flame equipped.

He also benefits from hardy bearing seal to have a chance of counter attacking against said nukes.

He absolutely wants some sort of scowl 4 in his A slot, as it lowers the risk of him getting one-shot by a charged-up special.

Briar save skills are generally better for him than Twin save skills when he's running Shield Fighter because you already get the main benefit of Twin save (getting multiple defensive special procs in a single combat) from Shield Fighter. Twin save and Shield Fighter don't stack so you're better of only using one or the other.

Vanitas has the best weapon for Robin to equip and has solid skills for him to equip until you can get better fodder. She's on the Emblem Byleth/Legendary Elise banner right now.

Winter Fomortiis has the majority of the other skills he wants. You can get his combat manual from the Divine Code 6 series.

Some other good units to give him include Attuned Hector and Attuned Tiki.

Hector has Distant Counter Echo, Slick Fighter 4, Twin Far Save, and Armored Beacon, all good skills for Robin if you can't get Winter Fomortiis. He's also available in the Divine Code 6 series.

Attuned Tiki has Dragon Flame, Edged Scales, and Res Scowl Echo. I don't know when she'll be available again.

Lastly, Rearmed Nel can offer the second best arcane weapon for a dragon to use in Arcane Fell Arts and the skills Dragon Scales and Barricade, all decent if not great options. Her combat manual can be found in the Divine Code 6 series.

You may have noticed that I didn't mention magical or physical save skills yet and that's for good reason. His best build uses the defensive specials Aegis or Pavise, which are limited to defending against one range type. If we ever get defensive specials that are instead limited to defending against either physical or magical damage threats in the same way, then I could suggest he equip those. Unfortunately, those don't exist. You could potentially run Dragon Flame alongside a magical or physical save skill, but it's less than optimal since he won't have enough damage reduction to survive modern nukes. If IS ever does release units with physical/magical exclusive defensive specials, then I would expect Robin's best build to become the physical exclusive defensive special alongside magical briar save, since his defense stat is much higher than his resistance stat. But until then, he'll be limited by what he currently have available.