Here are the system requirements for Final Cut Express:
System Requirements
A Mac computer with a 1.25GHz or faster PowerPC G4, PowerPC G5, Intel Core Duo, Intel Core 2 Duo, or Intel Xeon processor
For AVCHD: A Mac computer with an Intel processor
1GB of RAM or more
An AGP or PCI Express graphics card compatible with Quartz Extreme; or an Intel GMA integrated graphics processor in a MacBook or Mac mini computer
Some FxPlug filters are not compatible with integrated Intel graphics processors.
A display with 1024-by-768 resolution or higher
A DVD drive for installation
Mac OS X v10.4.10 or later
QuickTime 7.2 or later
500MB of disk space to install Final Cut Express and LiveType
Additional 500MB to install LiveType content
I have ran FCE on my iMac G5 2.1Ghz and 1.66 Intel Core Duo Mini. It ran just fine on both machines. Many things were faster on the Intel Mini than the 2.1Ghz iMac.
People who are poor have to make compromise sometimes. FCE will work fine on the Mini for now. No, it will not be as fast as on a Mac Pro 8-Core, but it will get the job done for now.