score:11
The "Ohio" in the name was about the Ohio river, not the state.
The background here is that when the Erie canal was built, it became a magnet for trade. A lot of western trade on the Ohio river used to be shipped overland to ports in places like Philadelphia and Baltimore, but now it was cheaper to haul it up to the great lakes, then through the new canal to New York City. Baltimore boosters needed their own solution, and there wasn't a convenient river.
The solution was to build one of these newfangled railroads, with the initial goal being to hook up with Ohio (and thus Mississippi) River trade by putting a terminus on "a suitable point on the Ohio River". By that standard, they reached their goal on January 1, 1853, when the railroad terminus reached Wheeling (at the time, part of Virginia).
Going into the new state of Ohio was a bit of a political problem. Not only was there a Civil War to wait out, but Ohioans (sensibly) wanted the trade to benefit their own railroad, not Baltimore's. So when a bridge across the Ohio river was built at Wheeling almost 20 years later, it was to connect their line to the Central Ohio lines (which the B&O by now had a part interest in).
Upvote:2
You've accepted an answer, but I thought I'd try to answer the literal interpretation of your question.
As T.E.D. stated, the railroad reached Wheeling, (later West) Virginia in 1853. However, there was no bridge across the Ohio River until after the Civil War. Competition with other railroads prevented the bridge from being built during the run-up to the war.
Following Sempaiscuba's comment, we find a timeline that tells us the railroad built such a bridge between Benwood, W.V., and Bellaire, Ohio from 1866 to 1871. They had to lease track on an existing Ohio railroad (and change the gauge).
(Source: Wikimedia Commons)
(Please note: As the Ford Motor Company was not manufacturing the Gran Torino in 1871, this photo is clearly from much later.)
This bridge still stands with its original stone piers across the river and stone viaduct on the Ohio side.
Also in 1871, the B&O opened another bridge from Parkersburg, West Virginia, and Belpre, Ohio.
So by 1871 the B&O was able to connect Baltimore with Sandusky, Ohio and eventually Chicago.