When you arrive at the entrance, ask a staff member that you want to go for nagashi somen.
They will give you a hand fan with a number on it (this will be your line number).
If the line is long enough, you will be asked to wait outside. Ask how long it's going to be, and if it's too long, it might be a good opportunity to visit some nearby temples and come back. That's actually what I did a few years ago (just do not wander for too long to not miss the time when your line number is called).
When the line is short enough, you will be asked to go downstairs to sit on one of the wooden platforms built right over the river - it is honestly one of the coolest lines that I had to wait in.
When it is finally your turn, they will once more show or call your number, and you will be shown where your seat is and which bamboo flume is yours. Each person/pair has their own bamboo pipe.
Get comfortable and ready to have some fun, and be sure to catch those noodles because if you miss, they will be gone for good (I always miss some 😂). Noodles will be coming down in small bite-size portions with some time in between, giving you enough time to: catch it, let the water drain a bit from the noodles, dip it in the sauce, eat it, and get ready for the next round.
I always forget to count, but I think there are around 8-10 little portions of noodles coming down the pipe, with the last one being pink, indicating that it is time to enjoy the mochi dessert and take some final photos if you want to.