Yes, an argument can be true even if the premise(s) is false. Validity is only about the structure of the argument, the form, which is why it's called formal logic. Having a false premise is irrelevant to the formal validity, and is instead an informal error.
As Mauro pointed out in his comment on your question, all conclusions can be shown to follow from a contradiction, so the example you gave is indeed a valid argument, although I don't think the example you gave is really that relevant to your real question. A more relevant example to your actual question would be an argument like this. "All bachelors are married. John is a bachelor. Therefore John is married." So just assume that john actually is a bachelor for this example. This is a completely valid argument, at least in terms of formal logic. Note, however, that even though it is a valid argument, it gives a false conclusion because it has a false premise (that all bachelors are married), and that premise will always be false.
Note also that it COULD give a true conclusion, if we had also made the other premise (that john is a bachelor) false, meaning that john really isn't a bachelor. Then the same argument would give the same conclusion, that john is married, but now the conclusion would be true, because the premise that john is a bachelor is false.
Basically it's just that validity determines whether that form of argument works and is completely consistent. If you take a false premise, then your knowledge about whether you'll get a true conclusion goes out the window (depending on how many premises are false and how many are true, etc).