People do say "I've run out of battery." as well as "My phone has run out of battery." It's run rather than ran in the present perfect tense (has + past participle of verb). Ran would be used in the past simple like "My phone ran out of battery yesterday, that's why I didn't call." You also don't need 'the' before battery.

It is possible to say" I've ran out of the battery" except WITHOUT the "the". :D But whatever you say, DON'T say "the" battery. You sound like you learned English from an online program, which is what you don't want :D

I would say My phone's dead, it needs to be charged. :}