For syrup, stir rhubarb, honey, water, and ginger together in a small saucepan. Bring to a boil; reduce heat to low and simmer, covered, for 15 minutes. Remove from heat; let cool completely, about 1 hour.

Strain through a fine-mesh sieve into an airtight container or jar; discard solids. Syrup can be stored in the refrigerator for up to 1 week.

For green tea punch, bring water to a boil in a medium saucepan. Remove from heat and add tea bags. Steep, covered, 5 minutes. Remove tea bags, or strain to remove loose tea. Let tea cool completely, about 1 hour.

Muddle mint in a large pitcher. Add cooled tea, vodka, honey-rhubarb syrup, and lime juice. Chill at least 30 minutes, or up to 12 hours.