The midfielder, who played every one of Swallows' games last season, is out for two weeks after damaging his right knee and ankle in the Telkom Knockout quarter-final loss to SuperSport United on Saturday.
Tsutsulupa was previously sidelined with a groin strain in the 2010/11 campaign.
"Lefa is going to be out for two weeks," Dube birds boss Zeca Marques tells KickOff.com.
"But the show goes on, it's part of the game. You are here today and tomorrow you're injured.
"The players will have to do the job, whoever comes in his position. Of course we have to change a few tactics, but this is what the game is about and we have to come up with solutions."
Marques is aware Celtic's players will be eager to impress their new coach Ernst Middendorp in the upcoming tie at Free State Stadium.
It will be Middendorp's first game in charge of the team following his departure from Maritzburg United last week.
"The players will be motivated to play against us," Marques says.
"Celtic is one of those teams that gives us difficulties.
"We are looking for the guys to find the back of the net quicker than what we did on the weekend.
"Every game from now is going to be battle for us."