Born in Rockhampton, Queensland, Munster played his junior rugby league for the North Knights at five-eighth, before being signed by the Central Queensland Capras in the Queensland Cup. In June 2013, he signed a 2-year contract with the Melbourne Storm. On 21 September 2013, he won the Queensland Cup Rookie of the Year award.
Munster played for the Storm's NYC team in the earlier rounds of 2014 before moving on to the Storm's Queensland Cup team, Eastern Suburbs Tigers. On 18 March, he extended his contract with the Storm until the end of 2017. On 3 May, he played for the Queensland Under-20s team against the New South Wales Under-20s team, playing off the interchange bench in the 30-8 loss at Penrith Stadium. In Round 12 of the 2014 NRL season, he made his NRL debut for the Storm against the North Queensland Cowboys.
In Round 6 against the Canberra Raiders, Munster returned for the Storm, filling in for the injured Billy Slater at fullback and also scoring his first NRL career try in the 14-10 win at Canberra Stadium. From Round 12 onwards after incumbent fullback Billy Slater was ruled out for the rest of the year due to a shoulder injury, Munster stepped up and filled in the vacant position at ease. In Round 22 against the Gold Coast Titans, he scored a hatrick in the Storm’s 36-14 win at AAMI Park. Munster finished his promising 2015 NRL season with him playing in 19 matches and scoring 7 tries for the Storm.
On 12 January, Munster was selected in the QAS Emerging Origin squad. On 31 January, while in the Emerging Maroons camp, he broke curfew and the QRL announced he would be excluded from its representative programs for the next 12 months. On 1 February, he was named in the Storm's 2016 Auckland Nines squad. He started the 2016 season playing at centre in round 1 against the Dragons. Following Billy Slater's season-ending shoulder injury, Munster took over the fullback role. He played fullback in Grand Final loss to the Cronulla-Sutherland Sharks.
With Billy Slater set to return Fullback and to action from a shoulder injury which ended his 2016 season (after round 1), Munster was expected to spend time playing Five-Eighth following Blake Green's departure to Manly, in the 2017 season. However with coach Craig Bellamy opted to rest Billy Slater for the opening 2 rounds of the season, Munster filled-in again at Fullback with Ryley Jacks playing Five-Eighth.
When fullback Billy Slater made his return from injury (coming off the bench in the 28th minute) in the Storm's round 3 home game against the Broncos, Munster then took over the Five-Eighth role. Following that round 3 game, he was then ruled out of action with a fracture cheekbone until round 7. He then took back the Five-Eighth role (combining in the team's spine combination with skipper Cameron Smith at Hooker, Cooper Cronk at Halfback and Billy Slater at Fullback), producing a number of great performances.
Munster was part of the Queensland Maroons Origin Squad for game 2 (of the 2017 State of Origin Series) as an 18th Man.
Following a season-ending shoulder injury to Johnathan Thurston, Munster was selected to make his State of Origin debut playing at Five-Eighth at Suncorp Stadium on July 12. On-debut, he produced an outstanding performance helping QLD to a series win (in the 22-6 game 3 decider win) over NSW.
* = Unfinished season