The main character who was a modern earthman (I think his name was Ivan) wanders into a medieval kingdom (I don't know if it was time travel or simply travel to another dimension). There he winds up married/engaged to a princess that he brings back to the modern world.
A big scene I remember clearly is when he first takes the princess to modern earth, she's naked as she doesn't have clothes in our realm/world/time. The MC offers her his shirt to cover up the obvious modesty issues but she refuses as it would violate the prohibition of crossdressing. She will not wear men's clothing. The MC then explains that things are different here and that it is much worse for a woman to walk around in the nude than it is to wear men's clothing, and that in fact as his wife/fiance/girlfriend most people would find it cute to see her wearing his clothes. They are also trying to hitch hike back to town.
I also remember that in the medieval setting he has a friend/ally who walks with a bad limp (I think he was also a monk). At one point the MC disguises himself as the guy with the limp and has to be coached on how to walk with limp. He fakes simply limping but that's not right as the guy with the limp has the limp but walks trying to minimize it.
Also in the end the MC and the princess have kids and are raising them jointly between the medieval world and the modern world.
I read this sometime between 2000-2005.