The cuts will affect 19 routes across Bristol and are expected to last until at least the beginning of April.
The announcement comes less than two weeks before the Clean Air Zone (CAZ) comes into force.
First Bus said it regretted the cuts, adding it was giving advanced notice "to make it easier for travel plans to be adjusted".
Axed journeys will show as cancelled on the First Bus app, as well as on the information screens at bus stops, First Bus said.
The company said in a statement on its website: "We are doing our utmost to address the driver shortage but doing so will inevitably take some time.
"As a result, we expect these cancellations to remain in place until at least the next major change to timetables on 2 April, which is a date agreed between all operators and the West of England Combined Authority."
It continued: "Cancelling these journeys in advance also enables our operations teams to focus on delivering a more reliable service for our customers, rather than dealing with cancelling journeys on the day."
The Bristol CAZ is due to start on 28 November and will see drivers of the majority of the most polluting cars charged £9 a day to enter the city centre.
The aim is to improve air quality, by encouraging people to either upgrade their vehicles, to take public transport or active travel such as walking or cycling.