This description can be a little bit confusing. What this achievement consists of is playing, within the same world in Minecraft, 100 days. You do not have to play Minecraft for 100 days (3+ months) to get this achievement. Simply play through 100 in-game days and this achievement will be unlocked.
You may obtain this achievement through play in any world where achievements are enabled. That means that creative mode worlds will not be eligible as those worlds turn off achievements.
As pointed out in the comments, sleeping through the night does not seem to help make the in game time go by faster. This achievement takes about 33 hours of real time. You can total the time through different worlds. No need to stay in the same one if you do not want to.