Yes it's equivalent, but the grammar is poor. Use a word like "the" or "your" before each noun here. For example: The assignment is now up on the blackboard. Your assignment is now available in the blackboard. The assignment is up on our blackboard for everyone see.

or "the assignment is uploaded onto blackboard."

Yes, you can say "The assignment is up on Blackboard," and it is equivalent to saying "The assignment is available on Blackboard." Both phrases mean the assignment has been posted and is accessible.

Are you talking about the educational technology system called Blackboard? If so, then it's most common to not have an article if it is this proper noun/commercial system. "The assignments for the semester are now up on Blackboard." "After I have graded the tests, you can find your results on Blackboard." "You can find the course syllabus as well as other supplemental files on Blackboard."