Is it possible to use Apache Solr to search across multiple sites?
Solr can be thought of as a document oriented database , it has a schema based, and a schema-less mode.

When used from the drupal module it uses a schema.xml file to specify what it expects to find in the documents (and hence index).

so if you go down that option one way you could search across the external domain and the drupal site is index the external domain to that schema, extending it where necessary

if you've got a title field in both sets of data , then it will index title that way. if your external site has different fields you could add them to the schema
you'll want to define a minimal shared set that documents from either site share, to make themeing, and UX easier.

The process I would probably go for is 


create a schema that covers both sources of data
configure Drupal to use that schema and index the site
index the external site (Solr 5 has a crawler now, or might just be easier to build a CSV file with curl)


that would get you to a place where you can search across the 2 sites from the Drupal search. 
It would be a case of getting the theme/markup right to represent offsite vs onsite results.