That's a hard to answer question. I am not saying Jesus would not use a cell phone, but don't you think he would use it a bit differently than we do?
First of all, for some people a cell phone or smart phone is an issue of power and prestige. It makes them feel important. It's like a remote in a hand of a man. Well, Jesus was not concerned about pride and power. In the words of Jesus, "For whoever exalts himself will be humbled, and whoever humbles himself will be exalted. (Matthew 23:12 NIV)" Jesus came in humility and as a servant. He didn't need a thing or an object to make him "important." So, no, Jesus would not use a cell phone to give him prestige or power; he wasn't interested in it.
Well, some people use a cell phone to keep track of their children or spouses or etc. They want to have access to them at all times. It's almost used like a leash to keep them in control, or at least, within reach. Well, Jesus probably would already know where they are. But even so, he would trust his Father to protect them. Of course, we might argue it is not a lack of trust in God as much as a lack of trust in the other person. Even so, Jesus sent out his rebellious and untrustworthy followers many times, and he had no gadget to keep track of them. He trusted that God could protect them and teach them through all their experiences even without his interference.
Other people use the cell phone like a play by play announcer in sports. They are always talking, and quite often they only communicate trivial nonsense, "I just saw..." Well, Jesus probably would not use the phone like that either. The only words he spoke were words God told him- no words wasted. I'm sure Jesus liked silence and meditation, something we so much try to avoid. So, Jesus probably wouldn't use it for that purpose.
Some carry a cell phone out of fear, "What if this happens or that happens?" The famous "What if..." We ourselves have a cell phone to be used basically for emergencies. Well, "what if's" are based on anxiety and fear. Jesus did not live by fear but with peace, so he wouldn't use a phone based on fearfulness. Of course, this goes back to trust- trusting in God.
Would Jesus have access to the Internet on his cell phone? I don't know. But remember, Jesus wasn't interested in information, but truth and wisdom. Much of the Internet is not truth. Knowledge of a fact is not wisdom but applying truth to life is wisdom. In fact, Jesus himself is truth, for he said, "I am the way and the truth and the life..." He wants us to know him and his Father (John 17:3). Can the Internet on the phone, or not on the phone, help a person know Jesus? Possibly, but there again, don't mistake knowledge about someone for knowing a person in relationship.
Would Jesus use the cell phone because he wants instantaneous information or to speak to you at this very moment? Probably not. Jesus is eternal. His time frame is totally different. He doesn't mind waiting and he doesn't mind having others wait. Would he have called Martha and Mary about his two day delay (John 11), I doubt it. After all, time is relative to an eternal being.
Text messaging is extremely popular with people. It is great for sharing information, but it is hard to build relationships through it. Why? Because communication is 55% body language, 38% tone of voice, and only 7% word content. In the words of one person, text messaging "replaces a commitment to a conversation with a series of one-sided communiquès."1 Text messaging emphasizes only the words; body language and tone are just guesses which makes it more open to misunderstanding.
Well, what would Jesus use the cell phone for? Love. If the cell phone could be used to be used to build love relationships, then he would use it. He wouldn't need to use it to build a love relationship with God; no phone is needed, not even for us. But if there is a way to express "agape" love and build a relationship, he just may have used it. This may seem like a "pat" answer, but it is not intended to be. Jesus didn't rush around trying to do this and that. He didn't desire power; he wasn't fearful; he wasn't a gossip or a chatterbox; he was eternal and did not live in the instant; he had no spirit of rush; he sought to communicate well to have good relationships He was about building up people, so he spend time with them. And if, in some way, the cell phone could be used to build love relationships, just maybe he would use it, but not with all the baggage we have attached to it.
1Wired by Jerry Adler called, "Meet the First Digital Generation."