Usually when I’m arranging for the uke I try to spread things across the strings, campanella style, so you’re not playing the same string many times in succession.
But there are times when that doesn’t work out. It’s then I bust out the running man:
You pick the string (the E-string in the case) alternately with your index and middle finger. That dramatically increases the speed you can play things at.
Here’s what the technique looks like when you have to swap between strings:
When you really have to bust out the speed, you can bring your ring finger in for support.
The tab for that last clip is in the How to Play Ukuleles for Peace ebook.