Make the Strawberry Syrup: Puree the strawberries in a blender or food processor until smooth. Press the mixture through a fine-mesh sieve and measure out 1 cup of the strawberry puree. Add it, along with the sugar, to a small saucepan and bring to a light boil over medium-high heat, stirring until the sugar has dissolved. Remove from heat and allow to cool to room temperature. (The strawberry syrup can be kept in an airtight container in the refrigerator for up to a month.)
Make the Drink: In a cocktail shaker with ice, combine the Pimm's No. 1, lemon juice and 1 ounce of the strawberry syrup and shake. Strain into an ice-filled glass and top with the 7UP. Garnish with sliced cucumber, strawberries and a sprig of fresh mint.