What I'm suggesting is a rechargeable 6V battery that's charged overnight by the transformer. The battery can run the pump during the day when the solar cells are not giving sufficient power and be (slightly) recharged by them when they are getting a lot of sunlight. You don't need an Arduino for that.
Plug a time switch into your power socket and plug the transformer into that. Use the 6V to operate a relay. When the power is on (overnight) the relay pulls in and charges the battery through a diode. In the morning, the time switch shuts off the power to the transformer, the relay drops out and the battery is connected to the pump and solar cells.
I use a similar system for my emergency lighting, using a 12V car battery and a charger. Any time the mains fails, the charger stops holding the relay in and the emergency lights come on 'automagically'. Beats having to find a torch in the dark when the RCD trips for no apparent reason.