The Premier League team were outclassed by League Two Bradford in the first leg of their League Cup semi-final and were beaten 3-1.
Lambert has been under growing pressure since his side conceded 15 goals in three brutal defeats over the Christmas period.
He said: "I feel exactly the same way as every Aston Villa fan feels. We shouldn't have been beaten. We shouldn't have conceded the goals we did. We had enough chances to win two games. You've got to take them.
"I'm not going to hide behind it. The way we were beaten. The goals we conceded and the way they were. They were really poor. The only good thing is we have a chance to turn it around. It will be a different game at Villa Park.
"Everyone is embarrassed by it. There's no other way to look at it. The only good thing is that we have another tie at home. You cannot defend like that.
"The club has been in exactly the same position with experienced lads. It's a challenge I'm not going to be frightened by.
"I can see it in the fans' faces. I don't need to read or see anything. It's my job to rectify it and pull it up."