Can Brazilian government shutdown, just like America's?

I wish it could! 

But, actually, it is not really possible. In theory, yes, but never in reality.

Brazilian Federal government can issue emergency acts (Medidas Provisórias) to allow pre-budget expenses. (Well, in reality, it cannot, at least according to the law. But it has been getting away with it pretty well).

Also, there are provisions that allow for the execution of last year's budget in case a new budget is not in place.

And, finally, there is little political interest in not approving a budget, since congressmen usually include large expenses that benefit their constituencies.

Nevertheless, I would like to see the Federal government go "on strike" for a decade or so. Business would become much easier in Brazil.