Apart from MySQL what other database can Cubecart use? Friends' web host does not offer MySql Database and so we need an option..

According to the requirements on the front page only MySQL is supported.

As Stevie says, you will need MySQL. There are no alternatives for this.

Either find a host where you can have remote access MySQL databases (fairly difficult to find due to security) or upgrade to a host that offers MySQL.

To be fair, MySQL should be fairly standard on most Linux hosting packages.