Peter, Hello and Welcome to the Forum...
Although Tel Aviv (The Hill of Spring) is now celebrating its 100th year as a city (founded in 1909), Jews from Jaffa had settled in the area and farmers, both Arabs and Jews had planted crops there for many years. In 1842 Aaron Shlush, an Algerian born Jew who had originally settled in Jaffa, as did most of the Jews of that time, purchased a large plot of land in the area that now includes the Tel Aviv neighborhoods of Neve Tzedek (the Oasis of Justice) and the Kerem HaTemani (literally the Vineyard of the Yemenites but more commonly thought of as the Yemenite quarter), most of which then either sand dunes or under cultivation (worth noting that a good deal of the cultivated land was of vineyards). Neve Tzedek was first settled in 1887 and the Kerem HaTemani in 1904.
Most modern historians agree that the market developed somewhat spontaneously, at first as a cooperative venture that served as an outlet for local producers and that the first outdoor stands appeared somewhere between 1890 and 1895. By the time the city was actually founded the market was booming and the major source of comestibles for the residents of the new city.
As to first restaurants - that depends partly on one's definition of a restaurant. First of all, we'll have to leave Jaffa out of this discussion because that ancient port city has a history of public dining places dating back to about 1470 BCE when Pharaoh Thutmose III conquered the city and many of his troops settled there. It is also known that Jaffa had several stop-over inns for sailors and caravans and that those had dining facilities.
As to Tel Aviv itself – anyone's guess as to what was the first true restaurant. It is known that the first kiosk in the city opened in the Neve Tzedek quarter in 1902 but the only thing that could be sold there was soft drinks. As to "eateries", it is well established that by 1900 several street-side vendors were selling prepared foods in the Kerem HaTemani. To the best of my knowledge no record remains of any first true restaurant.
If serious research is in order best bet will be to search out the Tel Aviv archives and those are open to the public. The archives are located in the Tel Aviv City Hall (69 Ibn Gvirol Street, Room 534). Access if free but one must contact the archives at least 24 hours in advance by phoning the archivist,
Nelli Verzervesky at 03 5218290.
As to oldest still existing restaurants in the city, my guess would be that three tie for that honor:
Elimelech and Shmulik Cohen (both founded in 1936) and Batia (founded in 1941). You might be able to find the name of the gastropub you mention by contacting Janna Gur at the magazine Al HaShulchan (On The Table). I do, however, wonder about the use of the term gastropub with regard to an early Tel Aviv restaurant as the phrase was coined in the United Kingdom as recently as 1991.