Drupal 7 - How to show an upcoming event from site 1 on site 2?
You can use Shared Blocks, it has both drupal 6 and 7 stable releases: 

If you have a main site with blocks that you want to syndicate across several
other websites, you need this module!

Features:

Allows publishing and subscribing of Drupal blocks
Changes to published blocks are updated on sites with subscription blocks
Published blocks provides a JSON endpoint at sharedblocks/{module}/{delta}
Block subscriptions are updated on cron with an adjustable refresh interval