I received an answer from my friend:
It is possible to run an ATMega328 at anything between 0.1mA and 20mA for only its core (I'm using an estimation from memory here, but there abouts) using the internal RC oscillator speeds. Lower clock speed = less processing power, but also lower power use.
So that will very much depend on the complexity of your program and how quickly it should run through that complexity.
The OLED is harder, because Sparkfun cleverly doesn't really tell us which display they are using, but for 48x64pixel RGB OLED, I would expect average use to be in the 20mA range. But this is just a very big ball-park guess, as I am not a super-expert on the stuff (so I might be off by a decent margin still). As much as the end-average use, even if you know the peak use or any other statistic, because an OLED display's power drain depends on how many pixels you use and how many colours for each pixel.
The best thing is to just measure what the display uses in your situation over a good span of time. Several minutes to an hour.
Using the Atmel datasheet you can make some predictions if you know at what speed it runs and using 3.7V (probably nearest curve will be 3.3V, so round up a little at least). Those sections are usually in the Chapter 24 to 32 region, somewhere at the back, called "Electrical Characteristics".
Also use at least 20% margin in your battery capacity to run time calculations, so you don't get disappointed when the actual battery is a bit off and/or starts aging, or might be defined at 10mA drain for a 110mAh type, where you drain 30mA, which would decrease that usable capacity.