Nice Uke! I am a rank amateur, and I learned all this from reading and heading to the man-cave (for three weeks), but I will happily share what I know about making a banjo Uke sing proper.
First step is the head, and that looks like a synthetic one. While it is not totally straw hat and raccoon coat collegiate...it is EASY. From the picture (and this is hard without it in my hands) it looks loose.
I went to banjohangout (the banjo equivalent to the underground) and found a recommendation to turn the heads up until the make an "A" note when you tap them. This worked amazingly well on mine so it is where I start. I used a table top tuner (not a clip on, but it may work) and tightened until I got that "A". Now, every head is different, so here is what you are looking for. As you tighten (in a lug nut pattern and only a 1/4 turn or so at a time) the sound will get brighter and brighter- pluck the A string continually to verify this. It will eventually reach it's "sweet spot" and then go totally dead pretty quickly... meaning you need to back off back to the best spot. My synth head never needs re-tuning, my calfskin one needs it daily...hourly...minutely. Make sure you play it a little open and feel the head tension so you remember it (if it is a natural head) and you will be able to put it back by ear in the future.
Sorting the head is going to raise the action, so here is what I have gleamed from following the greatest set up man in the biz. MGM (I think), asking questions here, and having two MGM Ukes to look at when I do one. Others will chime in if I am wrong here, but I have done my homework and this is what I do.
First, on an old timer, I take the ruler out of my square and make sure my neck is true by laying it on the fingerboard. A little wiggle in the fret height is ok...we can save fret dressing for another day. I had to do all this to my dear old Maybell so I can walk you through that too if need be...anyway, lets assume the neck is true and the frets are level:
Now, fret the third for each string individually, with a strip of a business card laid across the first. I used a whole business card the first time, but now I cut a 1/4 inch strip and it works mo' bettah. The string should just BARELY grab the card stock. We are talking a little resistance to being pulled out. If it is high, you need to get the files out and go for the nut (DO IT SLOWLY). Hard to explain what it should feel like, but probably what it felt like when you pulled the card from your wallet, unless it is full of junk like mine
Sorting the head will definitely raise the action of the bridge, and per MGM's recommendation, you are looking for 2.5 to 2.75 mm at the 12th fret. I do two things here. First, I seat all the strings in the bridge completely and level so they don't pop out when I get my groove onwith a little filing at the top, then I measure. If it is high, I have a highly sophisticated piece of equipment bring it down- a piece of 200 grit sand paper taped to a sheet of glass. AGAIN- go slowly and evenly, and measure often. This is a pain as you are tightening and loosening strings all the time, but worth it.
Once that is all cop-acetic, measure the distance from the nut to the 12th fret. Your bridge should be precisely that distance from the 12th as well, set it, measure it, and make a little mark with a pencil so that when you nudge it rocking out, you can put it back without scrambling for a ruler.
OK- thats what I got on this...now it is time for a little 5'2" and banjo uke bliss. Congrats on the new acquisition and hope this helps
Dave