They are actually abbreviations of Latin terms. "Inst." is short for the Latin "instante mense" or "this month." And "ult." is short for "ultimo mense" or "previous month." (Another related term genealogists may encounter is "prox.," from "proximo mense" or "next month.)
Oftentimes, an old newspaper item will report a total list of shipping arrivals and departures over a two week or 30 day period. These shortened terms replace the need to keep mentioning the actual name of the month.
"Inst. and ult." are used in many descriptions besides shipping news, however.
Here are two examples from 1870 newspapers. The terms "inst." and "ult." are highlighted in yellow. Please click on the images for a larger, more readable version.
Inst. A collection of news items for various towns in Ireland, 1870
|Irish American Weekly, NY NY 29 Oct 1870|
Ult. A list of recent deaths, from the Springfield, Mass. newspaper
|Springfield Republican 8 Apr 1870|
In 1923, Punch, a popular British satire magazine, referred to these terms as "outdated jargon."