Can I turn the Arduino power on and off using a transistor?
I think relay is good, but I want a smaller size.
Can transistor act as a switch between battery and Arduino?
Even if it's not transistor, I think other things of similar size will be fine.
You can control the power to an Arduino using a transistor (or mosfet).
How do you intend to operate (switch) this transistor on ?
Do you want it to latch on when it is switched ?
To help understand your requirements, explain why a normal on/off switch will not work for you.