The mercurial midfielder was substituted in the second half of Ahli's 3-0 victory over second-division side Nabaroh in the competition's round of 32 after picking up an injury.
"Barakat has no problems, he will play against Zamalek," Al-Badri told reporters after the game.
The 33-year-old played a key role in helping Ahli wrap up a sixth consecutive Egyptian Premier League title last month.
The Red Devils will miss one influential player in Emad Meteb when they meet arch rivals Zamalek.
The Egypt striker, Ahli's top scorer in the league with nine goals, is suffering from a niggling back complaint.
In-form and versatile midfielder Ahmed Fathi remains doubtful for the tie due to a groin injury.
Ahli are bidding to complete the double and win their 36th cup title in history.
They last won Egypt's oldest competition in 2007 after defeating Zamalek 4-3 in a thrilling game.