No, you cannot connect a VGA cable directly to an Xbox 360 Arcade.
The Xbox 360 Arcade model (the white matte console with 256MB/512MB internal memory) only has two video outputs:
It lacks the proprietary AV/HD port found on the original "Fat" Xbox 360 models (Pro/Premium/Elite) that supported the official Xbox 360 VGA HD AV Cable.
If your monitor only has VGA input:
This is the only way to get video on a VGA monitor from an Xbox 360 Arcade.