As an admin, if you have all rights granted to your account you will see the following options when you login to New UI,
Check whether you are having the following Navigate access rights to access the menus,
Project Management - Navigate access right
Staffing - Navigate access right.
Timesheet - Navigate access right - I see you have already activated Timesheet options under system options from Classic PPM.
Note: If you have activated the New Timesheet users will not able to access the Classic PPM Timesheet.
Administration:
Administration - Access
Blueprint - View - All
Blueprint - Edit - All
Blueprint - Delete - All
Blueprint - Create Copy
To Access Classic PPM from New UI:
Classic PPM - Navigate