Another question. This arcade will have Hyperspin and only MAME (no other consoles). Fry's has a computer for less than $200 that I believe should run it just fine. It has the AMD E1-6010 with integrated Radeon R2, am I right that this would be fine?
Someone who only plays the classics and uses an old version of MAME will tell you yes.
Someone who plays 3D games (Blitz, Tekken) and shmups from the 90's on a current MAME build will tell you no.
I don't have any experience with the processor you mentioned, but googled it and saw a spec of 1.35Ghz.
Newer processors do more with less Ghz, but I'd still be afraid of it falling way short.
Personally I won't use anything under 3Ghz anymore unless I'm just trying to find a use for an ancient PC.
EDIT: Try the games you want to play out on various other PCs you have access to and see if any of them choke.
It's all about where you draw the line. Ridge Racer plays wonderful in MAME, but requires a $200+ processor to run full speed.
I have 3 emulator systems in various capacities at the moment. They are all 3.4Gz Athlon X3 processors with used 9800GT video cards.
They cost around $250 in parts at the time, not including the OS.
They are just barely good enough to run the Tekken games in MAME and Naomi shmups in Makaron emulator. The Naomi shmups choke a little bit if run on Demul emulator.
PC games like Super Street Fighter IV and Mortal Kombat Komplete run surprisingly well at 1280x720 and medium settings.
This is back a few generations, so I wouldn't recommend building the same thing now, but it gives you an idea.
Current MAME builds are optimized for Intel processors.
If building today with the same budget, I'd probably go for a 2.8Ghz or higher i3.