What is the most number of hits allowed by a team in a nine-inning
Major League baseball game (post 1900) while surrendering exactly one
run?

Well, the record for most hits in a game without a run is 14.

http://www.baseballlibrary.com/baseballlibrary/features/experts/06_06_01.stm#1

The record for most hits by one team while being shut out is 14, set
once in the NL and once in the AL. On September 14, 1913, Chicago Cubs
ace Larry Cheney threw a complete-game shutout against New York,
despite being beaten up by the Giant hitters. The Giants had a hit in
every inning, but had only one runner reach third base. Cheney was
helped by three New Yorkers getting caught stealing and one being
thrown out trying to stretch a single into a double.

The Giants? dubious record was matched on July 10, 1928 by the
Cleveland Indians, who somehow could not score against journeyman
Washington pitcher Milt Gaston. Gaston, who two years earlier had
allowed 12 hits while shutting out the Red Sox, survived six leadoff
hits and five Indians reaching third base.

As far as most hits in a game while scoring 1 run, that's a question
for elias sports bureau or someone.  Good Luck!

If 14 hits is the record for no runs, I do know of a game where there
were 13 hits and one run scored.  On September 18, 2004 the Flordia
Marlins lost 8-1 against the Atlanta Braves.  They had 13 hits and
only one run.

actually it was on september 17, 2004.