Um, I was just in Japan and streets definitely have names. For example, this is in Nagasaki:
Meeting points and directions are generally expressed in terms of landmarks like train stations.
The system isn't really that strange. We (in america) use the same system of space decomposition for states, counties, and (in some places) cities. This just continues breaking it down to city regions (区), district name, chunk of blocks（丁目), block number, and finally house number. It's consistent with the larger scale system, and it handles change pretty gracefully.
In the block themselves there may also be a map of the block with the houses and the family names of the owners. Houses themselves don't have a number plaque but they have the owner's name at the entrance. Apartment buildings tend to have a (more or less pompous) name as well. Here "mansion" means an apartment :)
Generally you don't need to ask though, since the websites for japanese shops usually have a detailed map, and people who invite you will give you landmarks.
Also, to me the word "opposite" seems misplaced. I think that "complimentary" would be more apt. It follows that adage that whenever there is more than one way to do something someone will choose each.