Note: all links here are in Hebrew, except the first.
We have seen before that there was an unofficial Championship of Palestine in 1929. Tarbut Encyclopedia (Masada Press, 1964), in "Jews in Chess", adds that the the winner was Victor Winz, one of the strongest players in Palestine at the time.
This was not, it should be noted, the first tournament where players from various cities had joined. Already in 1922 there was a Tel Aviv - Jerusalem match; here is a report on the second (1923) tournament, which mentions in particular Pappo, the Tel Aviv player, as being successful (in Hebrew).
What's more, Marmorosh, in the mid-1920s, was promoting chess all over Palestine. While from Tel Aviv, he played for a while also in Jerusalem clubs, and even won the Jerusalem championship in 1928, while Pappo, playing for Tel Aviv in 1923, played in the Jerusalem championship in 1924.
As people did not "move about" too much at the time, it is likely that the clubs simply didn't care where a strong player lived -- as long as he joined and played. So the whole matter of which tournament was the first "general" or "nation-wide" tournament in Palestine depends more on definitions than on the facts. Was it in 1922, 1924, 1928 or 1929?