Modern Vespa

Got a 2014 LX 150, bought it recently, private sale. It was ridden regularly and apparently the guy had a friend (Lexus mechanic, he says) that maintained it regularly, but I don't have any records. It's got 2400 miles.

I just got new tires put on it and got it an oil change. I'm also going to get the drive belt replaced, though it's not in bad shape, just old.

I had it out for a mini spin today, and everything was fine in its performance. Just noticed a few things:

1. At a certain point, it got quite rattly or shaky. Not in steering or anything, but like there was a definite rumble from the engine, especially in idle. Thoughts on that?

2. When I got onto my driveway, which is sloped, I noticed that my rear brake was doing nothing. Nothing at all. I held the rear brake in, and I slid backwards down the driveway. Front brake works just fine. I'm going to take it to the mechanic, but is there a way I can tell what's wrong with it before I make the appointment? How can I check if it's fluid, or the brake line, or the pads needing to be replaced?

Thank you all!

Maybe check the cable