We actually spent very little cash on either of our two cruises. I bet the actual cash outlay was no more than $100, and we didn't take any travelers checks. Your actual cruise fees, port fees, taxes, etc. should be all paid for before you set sail.
For our 7 day cruise on the Sea (2 people) , we ended up with somewhere between $800 and $1000 on our onboard credit account. That amount included $10 pp per day for tips ($140), about $100 in internet fees (had to keep track of the kids at home), a soda card (around $30), shore excursions (between $500 and $600), miscellaneous non-alcoholic drinks ($4-$6 per drink with souvenier cup), and a couple of purchases from the gift shop. I don't know what happens if you gamble -- we don't do more than drop a couple of nickels in a slot machine.
I think that you can take cash (or probably travelers checks) down to the desk, and pay off the expenses as they're charged. What I don't know is if they have a minimum amount of credit to open your onboard account -- maybe if you call NCL they can tell you. You may also be able to put down a cash deposit at the beginning of the week, and have them work off that.
Obviously, your charges are going to be very dependent on excursions -- they were by far the biggest charges on either of our 2 cruises, outside the cruise fees. You should be able to book them in advance, and know what to plan for. Also, if you're into non-alcoholic or smoothie drinks, you might want to consider a "teen passport" if you can get them to sell you one. You get a book of 20 coupons for a little less then $40 -- way cheaper than buying the individual drinks.
