Yester-day, I listened to the Beatles' Sgt. Pepper's Lonely Hearts Club Band. For the first time, I realized how much the instrumentation does for "When I'm Sixty-Four." I had to look up exactly what instruments I was hearing though. According to Mark Lewisohn's The Official Story of the Abbey Road Years 1962-1970, there are two clarinets and a bass clarinet. I feel that those in particular help give the song it's old time-y feel.
In the time between that realization and looking up the specific instrumentation, I also realized that "When I'm Sixty-Four" has a connection to "P.S. I Love You" in that they both feature epistolary writing. "P.S. I Love You" starts out with "As I write this letter" and continues that theme through the lines "I'll be coming home again to you, love" (indicating the separation) and - of course - "P.S. I love you." "When I'm Sixty-Four" also uses that letter-writing language with the line "Yours sincerely, wasting away."
What I'm unsure of is whether that letter section of "When I'm Sixty-Four" pre-dates or post-dates "P.S. I Love You." Obviously "When I'm Sixty-Four" was released after "P.S. I Love You," but as Lewisohn's book notes, "["When I'm Sixty-Four"] was not a new song, the Beatles having performed a variation of it back in their pre-fame Cavern Club days whenever the amplifiers broke down." As Lewisohn mentions, that "pre-fame" version of "When I'm Sixty-Four" was probably different from the version that's on Sgt. Pepper, especially considering the Beatles' rapid evolution. So I don't know whether the letter section of "When I'm Sixty-Four" was written for the Sgt. Pepper version and was a throw back to that in "P.S. I Love You" (similar to how "All You Need Is Love" would reference "She Loves You" later in the year) or whether "When I'm Sixty-Four" was always that way and the letter elements in "P.S. I Love You" were something like a later try and an expanded version. And, of course, it could just be a coincidence that the two songs - both written mostly by Paul McCartney - use the same lyrical device.
In any case, I still think it's an interesting connection that ties the Beatles' more experimental period of recording to their earlier days.