The former Birmingham and Wigan boss is aware of the difficulties that Lambert will face as he hopes to keep the Canaries in the Premier League.
But Bruce is pleased for both club and manager after they secured a return to the top-flight with automatic promotion last season.
Bruce told Sky Sports News: “Paul Lambert has had a wonderful rise there.
"To go into Norwich and have back-to-back promotions is a wonderful achievement.
"Ian Culverhouse, his assistant - I saw him on the bench. He is as grey as me, but I played with him. He looked well.
"I am pleased for them, I am really, really pleased.
"I still keep in touch with one or two people down there. I can't quite believe it is 25 years ago - it's unbelievable - but I am very pleased for them."
The Black Cats boss enjoyed his time with Norwich playing in over 100 games and winning the League Cup.
However, the ex-Manchester United centre-back recalls his awful Norwich debut where he scored an early own goal against Liverpool.
He continued: “In my first game for Norwich, I scored an own goal after 30 seconds - it was against Liverpool at home - and I think I scored a second own goal as well about 22 minutes later.
"Kenny Dalglish cut inside Dave Watson - if you remember Dalglish's trick, his favourite one, he cut inside and bent it in the far corner with his left foot.
"I was on the line and I tried to get my head on that. I headed it and I headed that one in as well, so I had got two own goals after 28 minutes.
"I think they were calling for my head that day as well - 'Who's this clown we have bought here from Gillingham? Get him back to Gillingham'.
"Six months later, we were winning the cup against Sunderland, ironically, at Wembley.
"I had gone from the third division to playing in the first division, as it was then, so for that, I will always be grateful to Norwich.
"I had a wonderful three-and-a-half years. Even though it was a long, long time ago, I thoroughly enjoyed every minute of it.
"We won a cup, we got relegated, we got promoted, then finished fifth in the old first division, so it was a good time for me."