A timber-framed building at the junction of Northgate Street and Oak Lane, which is now used for offices.

This was one of three pubs of this name that have existed in the town, the others being on Tavern Street and Felixstowe Road (which itself closed early in 2014).

It's shown on this OS town plan of 1883.

There is some uncertainty about when this particular inn opened, as it isn't clear which historical records relate to this and which to the Tavern Street inn. However, we do have this document:

In Northgate Street, a corner house standing on the right hand side of the passage leading to the Church of St Mary at the Tower, was until lately known as the "Royal Oak." It is chiefly remarkable for its antiquated appearance, and a highly ornamental corner post, which represents on one of its faces a smith striking upon an anvil, and on the other a well executed carving of a man's bust, with elaborate carved work below, makes it specially interesting. Not so very long ago the house was occupied as a private dwelling, to which primitive state it has again returned. A house of far greater importance, bearing the same sign, was many years ago situate in Tavern Street, where it occupied part of the site upon which Mr. Corder's drapery establishment now stands.The old inns and taverns of Ipswich: their memories and associations, Evelyn White, C. H., Proceedings of the Suffolk Institute of Archaeology and Natural History, Volume IV, 1886

We also have a reference in the Ipswich Journal, August 2nd 1735***, to the Royal Oak in Brook Street, Ipswich. Unless there was yet another Royal Oak further south, this ought to refer to the Northgate Street inn, (but see caveat below).

This tells us that:

1. The Northgate Street inn closed fairly soon before 1886. (Another reference in the historical section shows this to have been late in 1882, so the license could be transferred to the new pub in Felixstowe Road).
2. White thinks the Northgate Street building was only an inn for a relatively short time. (How short though?)
3. Despite what White says, the Northgate Street inn seems to have been open by 1735.
4. The Tavern Street inn was the earlier of the two.

However, to throw a spanner in the works:

To be lett, the Royal Oak Publick House, near the Great White Horse in St Mary Tower Parish, Ipswich. The Stock and Utensils to be disposed of. Enquire at the abovesaid House or of Edward Reeve, in Brook-street.Ipswich Journal, April 23rd 1748***

So the reference in 1735 to Brook Street (the details of which are not currently available) might be talking about someone acting for the inn, perhaps a brewer?

As we have a concrete reference to "the Royal Oak Inn, situated in Upper Brook Street" in 1830, we can infer that:

1. The Northgate Street inn was almost certainly open by 1830 but may date back to 1735 or earlier.
2. The Tavern Street inn is certain to have been closed by 1830, but may have closed before 1735,

All of this assumes that what's now called Northgate Street was once considered part of Brook Street.