A bit of research confirms your guess, though we have to go back about 6000 years to find the common root!
The "bour" of "neighbour" is equivalent to the English word "bower" (an older term for a dwelling or shelter). Its closest cognate in Norwegian is "bur" (cage), both from Proto-Germanic "būraz" (chamber). Norwegian "bor" (reside) is from another Proto-Germanic word, "būwaną" (reside).
However, both these Proto-Germanic terms ultimately come from the Proto-Indo-European root "bheu" (be, exist, grow), hence the shared "bū-" syllable.
(sources: The American Heritage Dictionary of Indo-European Roots by Calvert Watkins, and etymonline.com)