If I am planning on buying a uke, I'd probably not play a whole song, but try out some strumming, some finger-style, some chords up the neck, a chord melody passage. . . and evealuate how the uke sounded and felt for each of these styles. Check for buzzes or high action, how comfortable the frets feel. . . It's always nice to have a friend who plays the uke come along so you can see how it sounds "out front".
Exactly my approach the whole TWO TIMES I've gone shopping in person.
I knew that I was going to be trying necks of varying radii, from flat to quite curved, so I did some extra practicing with barre chords on my current uke so that I'd have a better idea of whether a particular profile would be a better fit.
I also worked on memorizing a couple of passages that were giving me fits, along the lines of "Why I can't I get this? I'm doing just fine with other things that seem trickier, but this other thing is still kicking my ***.”
(I don't know why I typed it that way. I could have just said "kicking my asterisks." LOL)
The important thing for me was to mostly focus on what I do, with a lesser, but still specific, focus on the things I'd hope would be better.
As well as bringing along someone to play it for you, you might consider bringing your current uke to the store... or I guess "a" current uke from your collection that would serve as the best "compare and contrast" baseline if you have a bunch of 'em. I didn't do this myself, since I was previously playing a $249 mahogany laminate soprano as my only uke, and was shopping for a spruce-topped tenor in the $1200-$1600 range to replace it.
(I get very itchy when I have too many things around me, so I'm a fan of keeping myself down to one nice uke and a beater.)
So I knew what the baseline was, and I knew that more things were gonna be different than not.
But going forward, I definitely plan to bring the now current #1 as the baseline.