As you could see here, the Ben alias is a way for Obi-Wan to stay incognito on Tatooine. He moved there after the events of Revenge of the Sith which are 19 years before A New Hope. We could safely assume that Obi-Wan took the false identity of Ben as soon as he moved, so 19 years could count as "a long time".
Obi-Wan didnt tell Luke he "hasn't used "Obi-Wan" since a long time before Luke was born." here a quote from the Script of A New Hope
BEN: Obi-Wan Kenobi...Obi-Wan? Now thats a name I haven't heard in a
long time...a long time.
LUKE: I think my uncle knew him. He said he was dead.
BEN: Oh, he's not dead, not...not yet.
LUKE: You know him!
BEN: Well of course, of course I know him. He's me! I haven't gone by
the name Obi-Wan since oh, before you were born.
So he said he wasn't called Obi-Wan for a long time, and this time was before he was born. In fact, I checked the Revenge of the Sith script and he had been called Obi-Wan by Padme just after, but Yoda also called him "Master Kenobi". Anyway, "before you were born", that is a descent approximation for events that occurred 19 years ago.