Solr & User data
I'd go with a modified version of the first one - it'll keep user specific data that's not going to be used for search out of the index (although if you foresee a case where you want to search for favourite'd articles, it would probably be an interesting field to have in the index) for now. For just display purposes like in this case, I'd take all the id's returned from Solr, fetch them in one SQL statement from the database and then set the UI values depending on that. It's a fast and easy solution.
If you foresee that "search only in my fav'd articles" as a use case, I would try to get that information into the index as well (or other filter applications against whether a specific user has added the field as a favourite). I'd try to avoid indexing anything more than the user id that fav'd the article in that case.
Both solutions would however work, although the latter would require more code - and the required response from Solr could grow large if a large number of users fav's an article, so I'd try to avoid having to return a set of userid's if that's the case (many fav's for a single article).
Let's assume I am developing a service that provides a user with articles. Users can favourite articles and I am using Solr to store these articles for search purposes.
However, when the user adds an article to their favourites list, I would like to be able to figure out out which articles the user has added to favourites so that I can highlight the favourite button.
I am thinking of two approaches:

Fetch articles from Solr and then loop through each article to fetch the "favourite-status" of this article for this specific user from MySQL.
Whenever a user favourites an article, add this user's ID to a multi-valued column in Solr and check whether the ID of the current user is in this column or not.

I don't know the capacity of the multivalued column... and I also don't think the second approach would be a "good practice" (saving user-related data in index).
What other options do I have, if any? Is approach 2 a correct approach?