Premium processing for Fiscal Year 2020 cap-subject H-1B petitioners requesting a change of status (i.e. F-1 to H-1) will begin on May 20, 2019. If you did not request premium processing with the H-1b petition, you can still request it after May 20.
For all other FY2020 H-1B cap-subject petitions (i.e. consulate process), it will begin in June 2019 or later. USCIS now engages in a two-phased approach.
Premium processing for H-1B cap-exempt petitions, such as extension of stay requests, remains available.