If you're using RVM for 1.9.3, you can just do
rvmsudo ruby myapp.rb
If you've set up 1.9.3 some other way, figure out the location of your 1.9.3 binary by doing:
$ which ruby
/whatever/path/ruby-1.9.3-p125/bin/ruby
$ sudo /whatever/path/ruby-1.9.3-p125/bin/ruby myapp.rb
You could also just install Sinatra and any other gems needed for your system Ruby:
sudo gem install sinatra