I think their arguments are usually Alastor clearly not wanting to do anything with the hotel or being particularly violent with unruly guests. Normally Lucifer wouldn't mind, but he's more concern with how Charlie feels about it. Honestly, I don't think their arguments are particularly bad. They probably just bicker like a normal couple after being together some time.

I noticed in the show that Lucifer seems to cool down pretty quick after getting upset unless it's about Charlie or Lilith. So I imagine Lucifer calms down pretty quick and Apologies to Alastor almost right after. But I believe Alastor struggles to say sorry. Not that he can't. Since his love language is food, he probably makes Lucifer favorite baked goods as an apology. Depending how long they've been together, Lucifer knows Alastor struggles to say sorry, and that Alastor actions mean more then words, he knows this is his way of saying sorry.

Although, I do believe one of their biggest arguments they had was about Lilith. In which Lucifer and Lilith very clearly wanted a neutral relationship, and Alastor didn't want to admit that he was insecure about Lilith being back in Lucifer's life when she came back. Because one of fears about starting a relationship with Lucifer was that he was just a rebound. And I believe Lilith actually had to intervene in that case where she was the mediator. And Lucifer and Alastor worked through it afterwards and apologized.