How to attach maps, etc to handlebars of a Boris Bike? (London bike hire scheme)
Phone Attachment:
I would need to get better views of the front of the bike. Probably fab something starting with an existing mount (looks like you started down that direction).

Map Case: 
It looks like you already have a map case, if not, do a search. If you take one that will fit more or less in the gap on the bar and add some lengths of double sided velcro you should be in business. May require a bit of playing around to find the best attachment points.

I'd start with medium loops at the bottom circling the bars about where they attach to the "stem area" (not a real stem on the Boris bikes) and a bigger loop going around the front plate. If this tends to slide, you can add pieces of velcro at the back of the top strap coming down at an angle (think a 'K' rotated 90 degrees to the right) and securing underneath the map case.

Once you have it figured, you can wrap the velcro that is not actively sticking anywhere with some old jeans and run the sewing machine across. This will help keep it all clean and make it so you aren't spending all the time unsticking it from itself.

Searches:
'bike map case'
'velcro cable tie roll, 3/4" x 5 yards"'