What you are referring to is NOT Arnie's theme, but the WMEX "sign off" cart that they used to play when the station went off the air once a week on Sunday night at midnight for maintenance.
It was a recording of "I'll See You In My Dreams" (yes, you have the right song) that sounded like it was from the 40s or 50s. Then a female (!) voice came on over the music and signed off the station. I remember that she said "broadcasting with 5,000 magnificent watts".
It was a fun sign-off, and they used it for many, many years.