Hmmm... Lex would be there without money, without Luthorcorp/Lexcorp, and without any contacts. Probably just his prison clothes since in Supergirl land, he's in the joint. So, I don't know, he'd have to ingratiate himself with someone with assets he could co-op. Since Kara hasn't told Barry about Lex, it could be conceivable Star Labs takes him in as a refugee from another world. He feeds then some sob story so when Superman arrives, they have to go with the convenient plot device of wrestling with the question of turning Lex over to Superman. (Barry knows Kara has a cousin, Clark has heard about Barry from Kara, Cisco and Kaitlin don't trust Lex, but are hesitant about anyone as powerful as Clark Iris and her dad are against turning Lex over without a warrant.)
Meanwhile, as they are wrestling with their feet of clay, Superman, who easily outmatches this Earth's villains, is on a Super blitz, dragging down EVERYBODY. The League of Shadows/Assassins, Prometheus, The Legion of Doom (two guys a goon and a pizza joint), all of them. And, since they don't have kryptonite or magic, not much they can do about it.
Finally, they turn Lex over to him, and they go home. Ollie and Barry meet to decide what they need to do next, since every one of their villains are in prison. This opens things up to introduce new more costumed, less street clothes villains,
At least until the Crisis roles around and all Earths become one.