Should I use contextual filters to limit the view results for each type of administrators?
Contextual filters

I suggest you review the (free) video about Contextual filters with paths. It covers topics such as:


Using contextual filters for filtering Views result.
Using input from the view path to set contextual filter values.
Using contextual filter values for changing the view title.


If you like its content, you may also want to watch about 5 subsequent videos, also related to contextual filters (sometimes called "arguments" also).

Content Access

Even though your question is specific to Views, it seems like you should give it a thought to also get the Content Access module involved in your setup. Here is a quote from its project page:


  This module allows you to manage permissions for content types by role and author. It allows you to specify custom view, edit and delete permissions for each content type. Optionally you can enable per content access settings, so you can customize the access for each content node.


Just use this module to give appropriate access to each of the groups of users for each of the content types you have. After you do, the view you currently already have will probably not need any (or just a few) refinements. Because Content Access (if configured correctly) will make sure that the view results only include the news from each faculty.

Bonus: you'll find other use-cases for this module soon!

Organic Groups

Organic Groups is possibly an alternative also, which provides 'Groups' for Drupal users, and management of Access Control based on these groups.

In your case, each faculty would probably correspond to 1 such "group". And similar to using Content Access, the view you currently already have will probably not need any (or just a few) refinements.

Bonus: you'll find other use-cases for this module soon!