Is it legal to use a paper and a pen during Android: Netrunner play?
Taking notes is not permitted, no.
From the current Tournament Regulations (v3.2):

Taking Notes and Outside Material
Players cannot take notes or reference outside material or information during a tournament
round. However, players may reference official rule documents at any time or ask a judge for
clarification from official rule documents. Official rule documents include all rules documents
and inserts available on the Android: Netrunner page of our website, those found in an Android:
Netrunner product, or any portion thereof.

Edit: Under NISEI, this has been relaxed for Casual tier events.
From the current Organized Play Policies document (emphasis mine):

Taking or referencing notes referring to hidden information or strategy is not permitted at Competitive-level events. Players may reference official rules documents or card text at any time, or ask tournament staff for clarification. Players may take notes at Casual events.