Is the interior of an Instant Fortress larger on the inside than the outside like a Mage's Magneficient Mansion (which is what the item is based off of)?
Based on the description I'd have to say it doesnt give an indication if the interior of the Fortress is different to the exterior, but it lists the dimensions as "20 feet wide (and long I'd assume since its a uniform pillar shape design) and 30 feet high"
To me, that would mean it could consist of two or three floors, (20 ft x 20 ft) space per floor with a 15ft or 10 ft high ceiling depending how many floors it has. Understandably that isnt alot of room, but then again, given the protection the Fortress offers that is a reasonable compramise.
Nitpick: It lists the dimensions as "20 feet square and 30 feet high", so no assumption needed about the length.
Things you do need to assume (or simply adjudicate as a DM) are the number of floors and the existence of some sort of conveyance (ladder, stairs, levitation trap) to the battlement on top.
Due to the existence of the numerous attachments to the outside world ("arrow slits on all side") rather than a single portal like MMM, it would actually be difficult to work out how it could be extradimensional. The reason I think the spell MMM is required is due to the size changing nature of the cube, the command opened door and the fact that all the things that are required in side (a ladder, at minimum) exist inside without being crushed when the cube collapses.
Also, 400 sq feet per level isn't that small, especially considering the relative thinness of the walls (compared to stone ones). Stone walls would be a dozen or more feet thick. According to the equipment section, 100 HP of adamantium is only 2.5 inches thick.
Pembroke Castle, considered the one of the greatest keeps in Britain (from my limited quick internet research), had an interior floorspace of only 346 square feet, due to its 16 foot thick walls. Granted, it was 80 feet tall, so it could house more things.
Stone walls would be a dozen or more feet thick.
Why on earth would they need to be nearly that thick. I've been to Fort St. Augustine which stopped cannon balls cold and it isn't more than 6 feet thick.
I'm going off the information I was able to find. The thinnest keep walls I could find statistics for was 8 feet. Some were as thick as 18 feet. I said "a dozen" because it implies a number somewhere around 12 (9-14), rather than a specific numeral, which implies that specific number.
We are also talking about medieval keeps, not a fort built in 1740. (I assume you are referring to Fort Matanzas in St Augustine, Florida. Google returned no results for a "Fort St Augustine".) Technology, especially defensive technology greatly, advances a lot in 700 years. (Pembroke was built in 1093.) I'm going to have to trust you on the thickness, as I have not been there and could not find any corroborating evidence online. Either way, I gave the dimensions of a specific keep, which listed its specific wall thickness and diameter (i.e. Pembroke Castle) which fit very well with the interior size of the Instant Fortress.