Why do our clocks not go out of sync with the day over a year? Our clocks use 24 hours for a day, our planet takes 24 hours to rotate completely and our planet revolves around the sun in 365 days.
Imagine at the start of the year, the side of Earth facing Sol is 'Side A' and the dark side is 'Side B'. The side of the sun that Earth sees is 'Side A' and the opposite side is 'Side B'.
After half a year, wouldn't 'Side B' of Earth be facing 'Side B' of Sol, and thus the day/night cycle would be out of sync by 12 hours?
I've been trying to get my head around this problem for ages and I can't seem to figure it out. It seems like the angle that the sun hits the Earth must change by about a degree per day until it's gone around 360 degrees. Would that not mean the clocks offset by 1/365th of 24 hours every day over a year?
You are correct, and in fact the clocks are offset. But most people never notice this because of the way days are defined.

If you measure the "day" length by the stars (instead of the sun) you will see it's not 24 hours at all. It's around 23 hours 56 minutes. (Check this by going out at night, picking a bright star, and noting the exact time it passes due south. Then observe it the following night and note it passes 4 minutes earlier.)  By the way this shortened "day" is the sidereal day.

Each day, the Earth shifts along in its orbit (by about a degree, as you correctly observed in the question). So if you go out and point at the Sun at Noon every day, you are pointing in a slightly different direction. (I'll also neglect the Solar Analemma at this point too as it's just too confusing.)

Really, the "day" of 24 hours is not related to the true rotation period of Earth at all; loosely speaking it's just the average time between sunrises, or sunsets, or noons.