The Tigers suffered only their second home defeat of the season when Manchester United fought back from 2-0 down to win 3-2 on Boxing Day and Bruce knows they cannot afford to slip up again against the struggling Cottagers.
"After Boxing Day and Manchester United coming to town, it's after the Lord Mayor's show, that's always a worry," said Bruce.
"It's back to Fulham and that's arguably the bigger game for us. We must have the mental capacity to play two games so quickly and get something from the second game.
"We need to pick up some points from the teams down below us. It's alright getting plaudits for playing well against teams like Manchester United, Chelsea, Manchester City but we have to perform on Saturday.
"When the big teams come here, the expectation is that they will come and beat us, but I don't think Fulham will be playing four up front like United did at the end. We'll have to find another way to get something."
Hull are currently in 12th place in the table with 20 points from 18 matches, while Fulham, fresh from a 2-1 win at Norwich, are in 18th place with 16 points.
Bruce added: "We need to make sure the gap we have stays that way. We're ahead of the relegation areas and the challenge is to maintain that over the coming months.
"If we can beat Fulham, it will make a hell of a difference."