(From young to old) – никому прохода не даст!
They are talking about a ladykiller's taste in women covering a broad spectrum from young to old.
I wonder if this phrase means something along the lines of "no one slips under his radar"?
"Не давать прохода" literally means that wherever you go, there's a crowd gathering around you, so you can't go through. Because you are so popular, etc. It's usually exaggerated, for example, if you're a pretty girl every man is trying to hit on, people may say: "Ей мужики не дают прохода." - although, literally, that's not true, you could just ignore them and go where you need. I don't remember seeing it used the other way around (e. g. a single person "не даёт прохода" to multiple other persons), but I guess that's a legitimate usage as well.
Your translation "no one slips under his radar" is very close, if you assume that "no one" are all women.
Никому прохода не даст / никому прохода не даёт - He keeps bugging / harassing everyone (implied every girl)
Another idiom with a similar meaning:
Ни одной юбки не пропустит - He won't miss a single skirt.