I LOVE the tactile dome. To answer your question, there is a fair amount of crawling, but to be fair...if you really went straight through without stopping, it would take you less than 10 min. Within that time there is some climbing as well (I'm purposely being cryptic, because I don't want to ruin it). You could probably crab walk or scoot through the majority of the crawling sections if you wanted.
By the way, you absolutely need to make reservations. It's highly unlikely that you can "walk on" to it. I would say go for it; it's really something special that shouldn't be missed.