Today, the Muslim shrine described by Robinson is surrounded by the Shiite village named Nabi Yoûnis and bears little resemblance to the drawing above. Also, the dome is now green.
The Muslim shrine occupies the site of an earlier Byzantine church which was apparently destroyed by earthquake. Some remains from this church can be seen in reuse inside the shrine. During the Mamluk period, the structure was rebuilt and converted into a Muslim shrine.
I have no way for evaluating whether or not this tradition is historically accurate, that Nabi Yoûnis is the place where Jonah was spit out. It is interesting to note that according to 2 Kings 14:25, a prophet named Jonah son of Amittai lived during Jeroboam II’s reign. This verse explains that Jonah announced large territorial gains for the kingdom of Israel in the time of Jeroboam II. For a brief moment in history, the boundary of the kingdom extended north to Lebo-Hamath, identified with modern Labwe in Lebanon. The Aramean kingdoms of Damascus and Hamath were also subjected to Israel. Nothing is said concerning the Phoenician coastal cities, so I do not know if Nabi Yoûnis would have been under some kind of Israelite control or not at this time as well.
Further note: a small side room in the Nabi Yoûnis shrine supposedly houses the tomb of Jonah. As with Noah, there are apparently multiple sites that are believed to be Jonah’s burial place. Another such tomb of Jonah is located in el-Meshhad, Israel, the site identified with Jonah’s hometown, Gath-hepher (see Picturesque Palestine 2: 61, illustration on 56).