I'm not familiar with AMD APU performance, so I can't give any input on that. I want to guess you would be fine, but don't take my word for it.
I put together a few builds for you because I enjoy doing it. I tried to stay in your original budget as much as I could and played around with different ideas.
NOTE: None of these include an optical drive or operating system.
Build 1:
http://pcpartpicker.com/p/2u4GCOn this first build I followed your Mini-ITX choice. I've never picked out an ITX PC before, so that was a first.
Case: I don't trust any case in this price range, so I just went with what you had. The included PSU saved some money.
Motherboard: Gigabyte has a pretty good hold on the Z77 chipset. The Z77 chipset is ideal for overclocking, which I encourage with your cpu cooler.
RAM: RAM is RAM. I went with 1333 MHz RAM because 2133 MHz RAM does not provide much of a benefit over standard 1333 or 1600 clocked RAM during general use. It is usually only visible in benchmarks and other specific situations. It's pretty much just for bragging rights.
CPU: I went with Intel in this price range because Osu! is more CPU intensive than GPU. Unless you are playing high graphics games, from what I've read, a Core i5 will serve you well in all aspects. The i5 3570K is unlocked for overclocking.
CPU Cooler: The GeminII M4 had nice reviews and was still low-profile for the small case.
Storage: I'm impartial between SSD's and Hard Disks, so I just kept what you had.
Final Price: $565. $30 more than your original list. The SSD's price is slightly unstable and may raise soon.
Build 2:
http://pcpartpicker.com/p/2u4HrI looked around MicroATX options to see what there was.
Case: Rosewill is a nice company and the case had nice reviews. It did not come with a PSU.
Motherboard: Asus is good at what it does. I still went with the Z77 chipset for overclocking purposes. Not much else to say here.
RAM: Same as Build 1.
CPU: Same as Build 1.
CPU Cooler: Same as Build 1.
Storage: Same as Build 1.
PSU: This PSU has wonderful reviews and is more than enough to support your system. Should be very reliable.
Final Price: $575. $35 more than original list. You could easily drop the PSU and find a case with a PSU included to save some money, if you'd like. As with Build 1, the SSD's price is slightly unstable.
Build 3:
http://pcpartpicker.com/p/2u4kJI went with an AMD CPU and tried including a GPU.
Case: Same as Build 2
Motherboard: A cheap board. The price leaves room for a GPU. I do not suggest overclocking with this board, as it will probably turn out poorly.
RAM: Same as Build 2
CPU: The FX-6300 is very good for the cost at this price range.
CPU Cooler: Did not include one. This motherboard should not be used for overclocking if you don't like your computer catching on fire.
GPU (Video Card): A very decent video card will help with high graphics and general low-graphics usage like Osu! as well.
Storage: If you go with a GPU, it's common to download games that will use it. Those games take space. Space is what this hard drive has.
PSU: Same as Build 2
Final Price: $545. $5 more than the original. The HDD's price is pretty stable.
If you have any questions, feel free to ask them. I'd be happy to learn if I don't know.