The Blues started their title defence with a 2-0 win over Carrick while Ards were beaten 6-3 by the Reds.
"Ards had a disappointing result so they will be eager to put that right - it will be tough for us," said Mulgrew.
"They gave us a hard match at their place last season so we know we have to be on top of our game."
The Bangor Fuels Arena encounter is part of a full Premiership programme on Tuesday night.
Crusaders began their bid for a third Gibson Cup success in four seasons by defeating Ballymena United 4-1 at the Showgrounds on Friday night.
Paul Heatley bagged a double and the winger is upbeat for a Seaview meeting with Warrenpoint Town, who lost 3-2 to Glenavon on their return to the top flight.
"Warrenpoint have done fantastic to come back up straight away," said Heatley.
"They will work hard, they'll be well drilled and disciplined and tactically they will have looked very closely at our team.
"But we have dynamism in our team and versatility with our players - will will go into the game with a lot of confidence."
Cliftronville also caught the eye at the weekend with Barry Gray enjoying a winning start to his managerial reign at Solitude.
The Reds will travel to face a Coleraine side missing the suspended Steven Douglas after his dismissal in Saturday's 3-1 win at Dungannon Swifts.
Dungannon keeper Andy Coleman was also sent-off in the game so he is banned for the visit to Glentoran.
Glenavon will hope to build on their win at Warrenpoint in the Mourneview Park game against Ballinamallard United.
Ballymena and Carrick will be eager to chalk up a first win of the campaign when they meet at Belfast Loughshore Arena.
Linfield captain Jamie Mulgrew says the champions will face an Ards side determined to bounce back from an opening day thumping at Cliftonville.