If you grab the flagpole with Tetris mode active you will be able to place tetrominoes during the cutscene. Since movement inputs are killed when the mode switches, it's most reliable to do this by screen-scrolling yourself into the flag. Using the screen scroll to clip big Mario into the block at the bottom of the flag will work.

This glitch alone will let you get a few blocks into the castle before you're supposed to. However, damaging Mario with tetronimoes during the cutscene also has interesting effects. When big Mario takes damage, he becomes invulnerable during the castle construction. When little Mario takes damage he disappears from the screen entirely as if it were a game-over, but the game-over trigger is delayed for some reason.