"I come on the first day of January."
Translation:Tagaim ar an gcéad lá d'Eanáir.
It can be very frustrating to learn this way, but think of this: if you persist with a language, sooner or later you leave the safe confines of the 'classroom' and enter the broader language 'world' where you will encounter many things that will only gradually make sense. Learning to live with that uncertainty is a very important skill and makes one a better learner.
Why would you use the preposition "de/d'" here instead of the genitive of Éanair? Come to think of it, under what circumstances in general do you use the genitive over "de/d'"?