Clemens and his party arrived in Jerusalem at noon on 23 September after departing the village of Lubban at 2:30 that morning. Riding ahead of their tents and equipment, they took rooms at the Mediterranean Hotel, “a large and commodious house, well situated near the British Consulate, and not far from the Damascus gate,” according to a contemporary guidebook. Clemens noted to himself: “Loafed all the afternoon in the Mediterranean Hotel.” But judging from the entries in his notebook, Clemens spent all of the following day inspecting the landmarks of the city (Murray, 1:73; N&J1, 432–35).

SLC to Mr. Esais, 24 Sept 1867, Jerusalem, Syria (UCCL 00149), n. 1.

Sam and companions reached Jerusalem at noon September 23, 1867. From here they took a two day side trip to Jericho and another to the Jordan River and the Dead Sea, camping at Mar Saba. The party departed Jerusalem September 29, 1867 at 3:00pm for Ramla. Mark Twain Project: Quaker City Itinerary.

There is, however, an unresolved issue of when Sam and Dan Slote actually arrived in Jerusalem.  I was unaware of this, not having read Paine's biography, but related in Scharnhorst's "Early Years":

He had been ill with cholera at Damascus, a light attack; but any attack of that dread disease is serious enough. He tells of this in the book, but he does not mention, either in the book or in his notes, the attack which Dan Slote had some days later. It remained for William F. Church, of the party, to relate that incident, for it was the kind of thing that Mark Twain was not likely to record, or even to remember. Doctor Church was a deacon with orthodox views and did not approve of Mark Twain; he thought him sinful, irreverent, profane.
"He was the worst man I ever knew," Church said; then he added, "And
the best."
What happened was this: At the end of a terrible day of heat, when the party had camped on the edge of a squalid Syrian village, Dan was taken suddenly ill. It was cholera, beyond doubt. Dan could not go on—he might never go on. The chances were that way. It was a serious matter all around. To wait with Dan meant to upset their travel schedule—it might mean to miss the ship. Consultation was held and a resolution passed (the pilgrims were always passing resolutions) to provide for Dan as well as possible, and leave him behind. Clemens, who had remained with Dan, suddenly appeared and said:
"Gentlemen, I understand that you are going to leave Dan Slote here alone. I'll be d–d if I do!"
And he didn't. He stayed there and brought Dan into Jerusalem, a few days late, but convalescent. 
As this is not mentioned in Innocents Abroad, the chronology of the Long Trip must be a bit inaccurate. I'm wondering where and when this actually occurred and when did Sam and Dan actually get to Jerusalem.

See Bædeker (1876) Route 3 Jerusalem page 144

See Bædeker (1898) Route 4 Jerusalem page 19

Murray Route 7 - Jerusalem page 76