The only way to progress the season pass is by completing the related objectives that award SP and, which are tied in to Ultimate Team, Clubs, Player Career and Manager Career modes.
To see the currently active objectives hit LT when you are in the season pass track and it will show what is outstanding in each mode for the week/season/bi-week. This won’t specify the Ultimate Team Objectives however, and to check them you will have to cycle through the objectives in the Ultimate Team mode (the ones awarding SP are usually in the “Live” or “Seasonal” sections).
In season one, the season pass has 40 ranks with each rank requiring 1000 SP (total of 40k SP to complete it).
As of writing this I am at level 34 with 24 days left on the season pass (having completed all the seasonal and most of the weekly objectives) so you likely do not have to complete every objective in a season to finish the pass but will need to do about 70% of them. I say this because you can get 250 SP daily from the daily objectives of playing 2 games and scoring 3 goals in Ultimate Team that can compensate for other objectives (assuming each season pass lasts 30 days that is 7500 SP from the daily challenges alone).