Are the Threads and barrel diameter identical on a Ruger 77 and a Number 1?

in a short answer, yes. they are both 1x16tpi. the contours are different and the #1 has an extractor cut. without going and looking at my notes i don't remember if the tenion lengths are the same

The lengths are different.

Some years ago I tried to get Ruger to make me a 77 with the 26" No. 1 barrel because I thought the threads were the same, Bill Jr. said it wasn't possible because of the extractor cut. Instead they found me one of four experimental 26" 77 barrels with the heavier .300 Win Mag contour that they had made with a 1-8 twist because of complaints that the 1-10 was not stabilizing heavy 7mm bullets, in testing their own 1-10 twist barrels and two of the 8 twist barrels they found that the 10 shot the heavy bullets fine and the 8 was no improvement. This particular 8 barrel was not used in the testing so they gave it to me. I only used 160s and 175s in it, it was fine, I have no idea how it would have done with lighter bullets.