I visited Jerusalem twice. I did not "experience the feeling that they were there with Jesus 2000years ago and think that they were one or another of his deciples". However, I did experience some sort of "deja-vu".
I entered Jerusalem (the Old City) through the Jaffa Gate. From that point on, I went STRAIGHT to the Wailing Wall, although I had never seen a map of the Old City before that day, and the Old City looks like a labirynth. It felt like something was pulling me towards the Wailing Wall. It was surprising to me and to the friends who took me there, who sort of kept running after me.
I do not belong to any religion - but what I felt in the Old City and mostly at the Wall, cannot be put into words of any kind, in any language. I did visit ALL the important sites (for all religions - not only in Jerusalem, but throughout the Holy Land). It was an enlightning experience that I will never forget. Hope that helps.