Israels can take one-day tour from Israel's Eilat to Jortdan's Petra. But that's all. As I know, Israels can't freely travel all over Jordan.
I can't help your friend to get in to Israel. He should pass by himself thru Jordan-Israel or Egypt-Israel border security. I can't invite him or give him recommendation, because I don't know him.
After your friend will pass the border – he can use public transport. Many people in Israel speaking Arab. Your friend may use app Moovit – this is GPS-based app will guide him to destination much comfortable way, using online data of any transport: bus, train, taxi or even bycicle.
Maybe he better to get a tour? This way automatically solve any problem with border security and sightseeing.
I can't escort your friend to Jerusalem. He surely want to go to Al-Aqsa Mosque (Jami' Al-Aqsa) or some other arab's holy place. But non-muslims are not allowed to enter there. To say more, it's just dangerous for me. Sorry.
Any way, I wish good luck to your friend. If he finally will get in, I wish him nice and relaxing traveling. Just he pass the border, he can be 100% sure of him safety and Israel's friendly welcome. I hope, he discover the "new" Israel for himself.