I'm pretty sure the faces are at least clear coated. I don't know about the barrels though. Sometimes those are uncoated aluminum but I suspect they are clear coated as well. That wheel cleaner is particularly aggressive. It may damage the black inserts. I think those are just painted though.
Unless you have really really aggressive brake pads, like track pads, you probably don't need that strong of a cleaner. They make that stuff for people who just want to spray it on and spray it off without using a brush. Every see BMWs and Jeeps driving around with those black wheels? Its for dissolving that level of dirt. People that let their wheels get that dirty aren't too concerned with some pitting of the metal anyways. They just want a quick spray and rinse product.
If you are concerned about the wheel cleaner you bought you could certainly just use any good carwash soap and a wheel brush. If you have stuck on brake dust there are a couple products I would recommend you look into.
One is Sonax Full Effect wheel cleaner. Its safe for pretty much any wheel out there. You need to rinse it off very very well. It stinks bad too! But it works great. Goes on neon yellow and turns purple as it dissolves the brake dust and iron contaminates. Try to keep it off your brake pads/rotors but some overspray is ok. Just rinse the heck out of everything.
http://www.detailedimage.com/SONAX-M49/Wheel-Cleaner-Full-Effect-P525/500-ml-S1/
The other is CarPro IronX. It works in a similar way to the Sonax but you can get it in a spray or gel. I like the gel for rubbing it into the tight areas around the spokes since I get a lot of brake pad iron filing stuck in those spots. The spray is just easier to use. This stuff smells bad too!
http://www.detailedimage.com/CarPro-M54/IronX-Iron-Remover-P600/500-ml-S1/
Other than that most of the time a quick bucket of suds with a good brush is all they need. Meguiars D101 All Purpose Cleaner is my goto wheel cleaner for everyday cleaning. Dilute it 4:1 and it cleans wheels very very well. It only comes in gallons but a gallon is $15 and will last you a few years. Dilute it 10:1 and its great for cleaning interiors. This stuff smells like cleaner and isn't too terrible like the other two.
(I used to use the D103 but they have discontinued it, this is the replacement but I have not used this new one yet)
http://www.detailedimage.com/Meguiars-M43/All-Purpose-Cleaner-D101-P378/128-oz-S1/
TL: DR - You'll most likely be fine using the stuff you have. Spray it on, agitate it a little with a soft brush, and spray it off. Just don't let it sit too long or dry on there and I bet you won't have any issues on the outer face of the wheel that you can see.
EDIT: Wheel brush. I have both sizes, I find the small size to be much easier to use. Personal preference though. Totally worth the price. I use it all the time.
http://www.detailedimage.com/DI-Brushes-M51/E-Z-Detail-Brush-P273/Mini-Size-S2/
