I imagine this depends on a lot of things, like how skilled the Romans were at cartography, whether this subject was included in the education of commoners or patricians, whether it would have been considered worthwhile knowledge, etc.
Would the average Roman have a sense of what the empire looked like on a map? Would they know of Italy's distinctive shape, or the outline of the Mediterranean, or of far off lands like the British Isles?
Or was this knowledge pretty much common only to military figures or administrators of the empire who would have had a lot of immediate use for it? Also, how common were maps in general? Have we uncovered any maps from the time period?
Probably not, at least according to the current scholarship. As Susan Mattern summarizes in Rome and the Enemy: Imperial Strategy in the Principate (Berkeley and Los Angeles: University of California Press, 1999):
In both geography and ethnology the Romans had concrete and practical information of a type suitable for short-term, tactical thinking--itineraries, catalogs of weapons. But on a broader scale their knowledge and understanding of the world around them was different in character; they pictured a schematic, oval-shaped world, framed by zones of bitter frost and scorching heat, surrounded by the ancient, impassable ocean, and inhabited at its edges by primitive, exotic, sometimes mythical peoples. (80)
Ancient sources in general show considerable ignorance about geographic space and scale, or what Mattern describes as "the difference between the one-dimensional 'odological' view of the world... and a two-dimensional cartographic one." The only surviving "map" from Roman times, the Tabula Peutingeriana (better image here), appears to be based on itineraries (such as the famous Antonine Itinerary) and is more useful for calculating distances than conceptualizing the world. It's hard to say whether it looked anything like the world map commissioned by M. Vipsanius Agrippa sometime before his death in 12 B.C. and displayed at the Porticus Vipsania in Rome, since we have few descriptions, though C. R. Whittaker has argued in Rome and Its Frontiers: The Dynamics of Empire (London and New York: Routledge, 2004) that it and essentially all other ancient maps also derived from itineraries. Mattern does observe, however, that some ancient cartographers did advocate a more realistic, "two-dimensional" image of the world (but apparently without much success).
I should point out that the current consensus among scholars is that Roman leadership did not think in terms of "grand strategy" or anything that would resemble modern strategic planning. Although they did know how to shuffle legions around and allocate manpower (possible using the aforementioned itineraries as mental references), they would often initiate military operations into foreign lands without adequate knowledge about the local conditions, topography, and peoples. Mattern's research suggests that the Roman nobility (the types of individuals whom the Emperor would call upon as advisers) generally appealed to emotion- or value-based judgements when it came to decision-making. :)