We are not told when Jonah went to Nineveh, but we can narrow it down. Jonah was a prophet to the northern tribes just prior to Amos during the first half of the 8th century:
Amos ( from the kingdom of Israel) and Jonah (from the kingdom of Judah) were contemporaries. Jonah only covers a short period of about 3 years and doesn't mention Judah or Israeli kings, but if we can date Amos we can date Jonah. Knowing the dates of the kings of Israel and Judah puts Amos' ministry from 755 to 763BC. In Amos 1:1 it says:
" The words of Amos, who was among the herdmen of Tekoa, which he saw concerning Israel in the days of Uzziah king of Judah, and in the days of Jeroboam the son of Joash king of Israel, two years before the earthquake. "
This earthquake occurred 755BC, and "two years before the earthquake" [it must have been memorable] makes it 757BC when Amos saw visions , so we know it can't be any earlier than 757BC.
Nineveh had a plague in 765BC, a solar eclipse in 763BC, and another plague in 759BC. These events meant they were expecting something frightful, so they believed Jonah's message when he came and repented: this would mean it can't be any later than 763BC.
This puts the date somewhere between 757 and 763 BC , so if you said Jonah went to Nineveh in 760BC you would be fairly close.