Bing may have been somewhat aware 0f Bowie by word of mouth from a relative or colleague, but I doubt Bowie would have been on the Pop culture radar of the 73 year old Crosby, esp. when Pop culture wasn’t as widely disseminated as it is today.
A month after the special featuring their duet, Crosby was dead of a heart attack. The special was broadcast on CBS about a month after his death.
"Peace on Earth," an original tune that Bowie sings an arrangement that weaved “Little Drummer Boy together was written in about 75 minutes, Bowie and Crosby nailed the performance with less than an hour of rehearsal.
It's unclear, however, whether Crosby had any idea who Bowie was.
If anyone has definitive clues as to what Bing thought of David Bowie, I would be interested in hearing your comments...