What do I need to do, when I have run out of credit for my cell phone?
In most of the world I think top up is well understood. Where I've been in Asia and Africa, the booth attendant or store clerk will also understand if you ask to add units.
"Top up" is known in the U.S., being the terminology used by MetroPCS, Boost Mobile, NET10, and of course Virgin Mobile. The major prepaid providers, however— T-Mobile, AT&T, Verizon Wireless— all use refill. Because pre-paid mobile plans are not as prevalent in the U.S., I would always specify that it is your phone whether you are topping up, refilling, adding credit, or adding minutes.
Note that in the U.S. one can top off, but this refers to filling something completely that was already mostly full, for example when fueling up your car.