I have some experience in this.
What it really depends on is what stick you've bought and how easy it is to muck around. Regardless, the first thing you need to do is ground yourself before reaching in there, simply by touching any metal. Unscrew the back panel, and have a look. If the cables are neat and allow easy access to all the parts it should be fine to work with.
Since the stick itself sounds wonky there could be 4 issues in my experience. The easiest to remedy is that the cable attaching the stick to the circuit board is loose, and simply needs to be pushed back into place. Electrical tape can be a solid option to hold in place, just ensure you use a very small amount. The next would be as recommended above screwing the gate a little tighter. Those would be the screws securing the platform of the stick to the rest of the case. Thirdly, there could be an issue with the wire itself, which would be an obvious fray and needs to be taped with elec tape. The last is the biggest problem, the circuit board itself is defunct, likely due to poor manufacturer soldering. That, unfortunately, is outside my area of expertise, and may require a replacement arcade stick.
Disclaimer: I am by no means an expert, as I have only worked on my own arcade stick, and taken a peek at some other modding.
That one has a super neat interior, so you should have next to no trouble replicating the steps above. If it`s under warranty, opening it voids that, so ensure it isn`t a problem that the warranty covers first.