Welcome
You'll have access to the phone app but you won't be able to purchase any upgrades through the app, e.g. Acceleration Boost for Long Range or Full Self Driving for either the LR or Performance. Acceleration Boost can't be bought at the time of order so I'm not sure how that can be included in the lease price unless you come to some sort of agreement with them post delivery. FSD can be added at the time of order so can be included in the lease cost.
The app becomes active a few hours after you take delivery of your car.
The Premium Connectivity is included for 1 year with both the Performance and LR. Lease users have found that this continues indefinitely as Tesla don't seem to have a facility to bill back the lease companies for this yet. Autopilot comes as standard on all models (this is lane keeping and traffic assisted cruise control).
When placing your order, if you don't already have one make sure you obtain a Tesla Referral code to get your 1000 free super charging miles, you just need to provide an order link to your lease company. You'll find that users on this forum display them on their profile page.
Good luck with the purchase, it's a great car!