I have a 3.7V 4.4Ah li-ion battery and would like to use it to power my Raspberry Pi (which needs 5V and 700+ mA). What’s the easiest way to do it with minimum EE experience? Also, what resulting running time can I expect?
The easiest way without having to do any EE experience would be to buy a boost converter module. You might probably get a little under 4.3 hours.
Once you've found a suitable boost converter a simple way to think about power consumption is to think in terms of watts used per hour. So 5 volts at 0.7A which the Raspberry Pi requires is 3.5W (just multiply together). For your battery 4.4Ah x 3.7V = 16.2 Wh, so 16.2 / 3.5 = around 4.6 hours of operation.
Then you'll have some loss in the converter, 80% efficiency is probably a good conservative value to use so 0.8 x 4.6 = 3.7 hours of operation. It won't be exact but will give you a good simple ballpark figure.