They're about equally common. There are many ways to express the idea. There isn't any stock phrase. It's just like the choices a person might make about hunger: "I'm hungry," "I could sure use a bite to eat," "I need some food," "I'm famished," "I'm starving," "A hamburger would hit the spot right now," etc. "Top up my phone," though common, is strange to me. I first heard the phrase "top up" in connection with fueling a car. It means "fill the tank to the top," put in as much gas as the tank will hold. "Fill the tank" means fill it, probably until the gas nozzle shuts off automatically. "Top it up" suggests really trying to put in every single drop that the tank will hold, perhaps pressing the trigger a second or third time after the first time it shuts off. The reason "top up a phone" sounds odd to me is that there's no way a phone can ever be "full" of airtime.

Yes, there a lot of different ways of saying this, including 'buy some air time', 'top up my phone', 'add some minutes', etc. I'd guess that 'top up my phone' is most common. If you've used up your allowance of texts/minutes/data then you can say 'My phone is out of credit' (not 'out of the credits'). We tend to think of "charge" in the context of mobile phones as being connected to the level of the battery. Therefore if someone says "I need to charge my phone." they normally mean that the battery is low, not the amount of credit. I will add a caveat: there may well be regional differences, or even different things said in different age/social groups so it will be interesting to see what other answers you get.

Depending on where you are, "add some minutes to my phone", "top up my phone", and even "add some talk time", etc, they are all acceptable. The only one that sounds really strange is "buy some air time". I've never heard people say that.

I would say 'top up my phone' or maybe 'add credit to my phone' or 'put some money on my phone' or something.. 'minutes' seems like you're going to say how many minutes you want.. and 'air time' seems like you're going to be on the radio.. I've never heard that (but it could be a translation of a natural expression in your language)