I am currently doing some research on the subject and hopefully can provide a more complete answer later, but apparently Capablanca did not invest much time. As he said (extracted from this compilation) about his tenth move:
I thought for a little while before playing this, knowing that I would be subjected thereafter to a terrific attack, all the lines of which would be of necessity familiar to my adversary. The lust of battle, however, had been aroused within me. I felt that my judgment and skill were being challenged. I decided that I was honor bound, so to speak, to take the pawn and accept the challenge, as my judgment told me that my position should then be defensible. – Jose Raul Capablanca
Edit 1: The tournament time controls on the late XIX and early XX centuries were similar (but not exactly the same) to the Classical time control, that is, 120 minutes for 30/40 moves, plus an additional 60 minutes for the next 15/20 moves (with obviously no time added back then, and occasionally 15 extra minutes were added for the rest of the game once the final control was reached). This is backed up by a few links to tournaments like Vienna 1898, Paris 1900 (1), Paris 1900 (2), Ostend 1907, St. Petersburg 1914 and Carlsbad 1929 (and I'll stop the counting here). There were surely many more with this 30 moves on 120 minutes plus 15 moves on 60 minutes or similar time controls.
Thus, I think it's reasonable to assume that Capablanca and Marshall were playing this time control. Now, the opening phase of the game is mainly an automatic one, where if still on book you simply blitz the lines you like. As my coach says, when you know a theoretical line, you don't waste much time thinking about the correctness of such line, you just play it out. Based on actual games from GMs, they invest around 15-30 seconds per move on the opening, that is, the first 10-15 moves (sometimes even more if they enter a big theoretical line). So we can safely assume that Capablanca's time had been reduced by more or less 5-10 minutes by the time he faced the decision to take or not the pawn on the tenth move.
If we assume this reasoning correct, I think that Capablanca had 110 minutes for the remaining 20 moves.
Edit 2: As the member batgirl of Chess.com pointed out to me, the information about the time controls for the tournament is available in the tournament book "International Masters' Tournament of the Manhattan Chess Club, October-November, 1918". According to it's 11th page, the rules governing the games at the event were:
THE CODE AND SUPPLEMENTARY RULES.
The following is a copy of the minutes of a meeting of the players and committeemen, held the day before play was scheduled to begin, and at which a number of special regulations were discussed and adopted to supplement the Revised Draft of the American Chess Code:
At a meeting of the competitors and committee (Messrs. Marshall, Morrison and Whitaker being absent), held at the Manhattan Chess Club on the afternoon of October 20, it was voted to adopt the Revised Draft of the American Chess Code (British Chess Code) as the Code to govern this tournament, insofar as its terms do not conflict with any of the special regulations adopted at this meeting and hereinafter mentioned.
After a discussion of the advisability of the adoption of the rule providing for the moving of the King as a penalty for illegal moves, it was voted to apply this rule (or "Penalty C" of the Code) only to such instances where the illegality affects sealed moves.
It was decided that a game may be declared drawn at any time by mutual consent of opponents.
Where any player sets up a claim (as for a draw because of a threefold repetition of position), which, upon investigation, turns out to be false, such player making false claim shall have charged against his playing time on his clock so much of the time as may be required for examination and demonstration of the claim, provided, however, he is not penalized more than thirty minutes for any one such false claim.
There shall be no analysis by any of the competitors of pending games, either their own or those of other competitors, during the regular sessions of play and the evening intermission, within the confines of the Manhattan Chess Club rooms.
The time limit shall be Thirty (30) moves in the first two hours and fifteen (15) moves each hour thereafter.
Any claim for exceeding the time limit must be made before the player, against whom the claim is directed, has completed his 30th, 45th, 60th, 75th (etc.) move. It shall be the duty of the Umpire to declare forfeited any game the moment a player has exceeded the time limit.
In the event of a player's withdrawing from the tournament, the games played by him up to that time shall stand as scored, if at the time of his withdrawal he shall have completed his schedule of games for the entire first round, in which case all of his other scheduled games shall be forfeited to his opponents; if he has not completed the first round (meaning half of the tournament), all of the games played by him shall be cancelled.
Adjourned games shall be played on Mondays in the order of the schedule. All adjourned games must be played to completion before any games of the second round (or half) of the tournament are begun.
The regular hours for play on six days of each week, exclusive of Monday, shall be from 2.30 P. M. to 6.30 P. M. and from 8 P. M. to 11 P. M. On "Mondays the same hours shall obtain for adjourned games.
Appeals from decisions of the umpire shall be submitted to the referee by the tournament committee of the Manhattan Chess Club and at the discretion of the said committee.
The entry fee shall be $20.00 for each player and shall be returned upon completion of his schedule of games, and forfeited if such schedule be not completed.
The prizes shall be: First, $300; second, $200; third, $100; fourth, $50. Non-prize winners shall receive payment at the rate of $5 for each point scored.
Prizes for the most brilliant games: First, $30; second, $20.
Thus, there were 60 minutes added to the clock for every 15 extra moves on the board after the 30th one.