Hey, in The Original Series, The Animated Series and the classic movies, the Enterprise went beyond the rim of the galaxy twice, visited the centre twice and made it back to Earth inbetween. These movies are just using them as the measuring stick, instead of that 75-years-to-cross-the-galaxy slowpoke USS Voyager.
If the new Enterprise has transwarp, the old one must've had it too!
Also: Hi and welcome, mymatty
! I'm sure there is some old episode where Kirk or Spock have said "X% of our galaxy is still unexplored" or words to that effect, but I can't think of anything now. I'd guess it's pretty much the same amount in the alternate universe as the original.