Direct translations that would not sound old Tagalog would be:
1.Bukas ang pasahan ng assignment (sounds like an old tagalog speaker)
- We don't usually use "takdang aralin", but you can definitely use it instead of "assignment".
- Tomorrow is the deadline of the assignment (kind of like stating facts only)
2. Bukas na yung pasahan ng takdang aralin/assignment (this one sounds normal to me)
-Tomorrow IS the deadline of the assignment (NA here is used to stress that tomorrow is the DEADLINE of the assignment, it's like when you stress a word when you are speaking.)
Bukas (tomorrow) / na ang pasahan (is the deadline) / ng takdang aralin (of the assignment)
Deadline /Duedate / submission just means "araw ng pasahan" or we just say "pasahan"
Just like Shakespearean english, native speakers here in PH don't use deep/old tagalog words anymore.
I'm not really good at English but I can definitely understand and translate it directly into tagalog (I grew up and still living here in PH for 28 years.)
Let me know if you have any questions!