McLeish moved to Villa Park from bitter rivals Birmingham City, but lasted just the 2011-12 season before being sacked upon Villa narrowly avoiding relegation from the Premier League.
Following his controversial switch from Birmingham, Villa's fans never took to McLeish.
Lambert's subsequent arrival was met with initial optimism, but he has since struggled to meet expectations, guiding the club to successive 15th-placed finishes.
"I knew it was going to be a tough ask for Paul," McLeish told The Mirror.
"I saw some of his early comments and I think he thought it [the fans' displeasure] would turn overnight. But I knew there was a wee bit of work to do.
"In the season I was there, we got the wages down pretty substantially but no fan is ever going to thank the manager for saving money. What they want to see is production on the pitch.
"But there is no chance that anyone can turn the project in a year, especially when you are reducing finances as opposed to giving the manager money to spend.
"There has had to be a bit of ducking and diving at Villa to remain competitive and to try to get up the Premier League. Villa fans don't like to see their team near the foot of the table but it is still a massive battle."
McLeish admires Lambert's work in balancing the books, but knows continuing to cut costs will not see discontent in the stands cease.
"Paul has done well also to get the wages down to a level which [owner] Randy [Lerner] felt was sustainable for Aston Villa," McLeish added.
"But if you are not getting results you won't get patient fans, I promise you. At any club, when the team is not winning they are vociferous.
"The bottom line is results and that is what they judge the managers on. But generally it is the guys who spend the money that dictate where they are going to finish in the table."
Aston Villa open their new league campaign at Stoke City on Saturday.
