とりあえず is used when you do/say something easily done/said before getting to a main difficult part.
Before my work, I had a cup of coffee first.
It's hard to explain, but I think it's incorrect anyway.
Putting aside what to watch, let's go to the theater first.
I did it as a quick fix.
At least, (I can say this now,) I'm relieved.
The use of とりあえず in your video is tricky and atypical, but it comes off to me as "it's not a big deal (to me)" or "I experience this very commonly", as if "its falling" were something he(?) experiences thinking とりあえず. It explains the following exchange, too:
So it... falls every now and then?
"Toriaezu", you say?! It falls off that easily?!
"I guess" is certainly not a dictionary translation, but I cannot think of a better translation...