I don't think so. The player could name it to give a clue, or [best practice] have a MOTD arena message pop up when you enter, which will list the major settings. There's been a suggestion for a custom arena BBS forum for advertising your custom arena; where you could list the day and time the arena will be available along with the settings.
Though I haven't tested it, I would imagine only the Host who created the virtual arena has access to the arena settings
Another wish on the wish list is that all arenas allow players to inspect that arenas settings via a disabled menu (greyed out - can read / cannot edit)
No reason at all to not allow a player to browse through the Arena Setup and inspect what the settings are...even in the MA's
Would be great for special events to make sure that the Arena Setup is indeed set correctly per a write up as other sets of eyes can inspect the set up and inform the events host of any discrepancies such as object hardness, downtime etc
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Oneway