This is a premium feature and requires the Standard plan or above.
You can customize the delivery time shown to your customers based on their location.
Location specific delivery times are based off of Shopify Markets, not your customer's actual location. You can only set custom delivery times for markets that you're selling in.
You will need something like the Shopify Geolocations app that allows your customer to select their preferred market for your store.
Customers can select their preferred market with the Geolocations app
From the Arrives By app go to the Locations section. Here you can see all the available markets on your store and any custom delivery times they have.
Locations are based on your store's active markets
Select the a market to edit its delivery settings. Note that you must set the status to "Active" to make it live on your store.
Sync Markets
If you add a new market and you don't see it on the Locations page then press "Sync markets". This will bring everything up to date.