can homemade arduino be powered up with battery?
First, of course, you'll need a battery with enough capacity, and voltage enough greater than 5v (assuming you want to run your home-built at 5v - typical, but you could run it lower, say 3.3v) to give enough run-time before the battery discharges to where it can no longer supply the system. Note that the 9v, so-called "Transistor battery" is not capable of supplying much current. A battery pack of off-the-shelf alkaline batteries or rechargables, or a Lithium Polymer rechargable battery will do much better.

You'll need a low dropout voltage regulator to reduce the battery voltage to your chosen Arduino supply voltage. National Semiconductor's LM1117 would be a good choice. The diagram below from NS's datasheet shows a simple-to-make voltage regulator. Its input is your battery; its output will be the supply voltage to the chip and any accessory circuitry.