In my experience you pay your bills in cash. You get your cash from the ATM machine, often with difficulty and high fees(more posts to this site on this problem than any other). You can open a bank account there but the problem is that there is no easy way to bring money into the country to put in the account, therefor having a local account is not much use unless you have local income.
It's not unusual to encounter long lines at ATM's or to have to go to multiple ATM's before you find one that works. Also be aware that on weekends many run out of money and that holiday weekends the problem is worse. I was lucky and had a Citibank account so I could go to their local branch about 25 minute drive in each direction and withdrawal up to $1000 at a go(make sure you have a Citibank or similar card, see the threads on this subject, very important).
One you have your cash you will be able to pay your bills, as Igor states you can use pagofacil, but in my experience this covers only some of the bills that had to be paid each month. Typically for us to pay our bills we had to go to about five different places each month. It's not unusual that in each place you have to stand in line to pay. Some waits can be quite long, over an hour was not uncommon. Taking a book or other reading material helps.
Depending on how many monthly bills you have you can count on spending the better part of a day and often more to take care of the basics. As it sounds we found this to be a major chore each month. I really don't know how Argentines who work full-time manage to get it done.
