Yes, you can place cash on your onboard card. You can do this through guest services once onboard. You will need to visit guest services again on the final morning for a refund of any cash you do not spend.
If travelling with children MSC also offers "My Teen Card" a pre-paid credit card for onboard purchases with a set spending limit. These can also be purchased with cash once onboard. No refunds on these though, if you don't spend it you lose it.