Monschein began his footballing career with hometown club SC Brunn am Gebirge.[3] In 2014, he moved to ASK Ebreichsdorf. In January 2016 he joined his first professional club Admira Wacker Mödling.[4] He made his Austrian Football Bundesliga debut on 7 February 2016 in a 2–1 loss to Red Bull Salzburg, when he came off the bench in closing stages for Lukas Grozurek.[5]
Austria Wien
On 1 July 2017, Monschein joined Austria Wien for a reported fee of €600,000.[6] He signed a three-year contract. During his time in Vienna he made 116 Bundesliga appearances, scoring 36 goals.[7]
LASK
In the summer of 2021, he joined LASK on a three-year contract.[8]
On 27 January 2022, Monschein was loaned to Rheindorf Altach until the end of the season.[9]
SV Ried
Monschein joined SV Ried on 13 June 2022, signing a two-year contract with an option for an additional year.[10]
First Vienna
In summer 2023, Monschein signed with First Vienna FC, on a two-year contract.[11]