Use Low Power Arduino with Solar Panel
You have far too many variables and unknowns there. Primarily you need to know what the average current draw for your circuit is. Secondly you need to decide how long the solar powers are allowed to take to recharge the battery. You have to think not only about runtime but also charge time. The charge time dictates the maximum capacity of the battery, and that then dictates the runtime of the Arduino. So here's some napkin maths to illustrate:

Assuming you want the battery to charge from almost flat to full within 8 hours, and with a 1.1W 6V solar cell, that gives you a peak of 183mA. 

Assuming the Arduino circuit takes 83mA on average (let's give ourselves nice round numbers here) that leaves 100mA peak for charging the battery.

If you're charging at, say, 0.2C, that gives a full charge in 5 hours. 100mA / 0.2C = 500mAh capacity (at peak sunlight and current). 

At 500mAh capacity and 83mA average current draw that would give you (0.5/0.083) 6 hours of runtime without the solar panels.

So that's just not going to work.  It'll be dead by morning.

Now assume that you can turn off everything for 99% of the time.  That effectively reduces our average current draw to 0.083*0.01 = 830µA. Let's plug that into the same reasoning as above.


Current draw = 0.00083A
Incoming peak current = 0.183
Available charge current = 0.183-0.00083 = 0.18217
Charge at 0.2C = 910mAh. 
1Ah battery would charge in about 5.5 hours of peak sunlight.
1Ah would last about 1200 hours, or 100 days.


None of this takes into account varying light levels or light incidence angles. As a rule of thumb, take the light as averaging 50% over the course of the 8 hour period. You will only get 1.1W when the angle of incidence is 90° to the solar panel. At all other times it will be reduced. So if we reduce the current from 183mA to, say, 100mA, the charge capacity changes to:


Incoming peak current = 0.100
Available charge current = 0.100-0.00083 = 0.099
Charge at 0.2C = 496mAh
A 500mAh battery would charge in about 5 hours.
A 500mAh battery would run for about 50 days at 830µA average current.


So you see everything really hinges around the current consumption of the Arduino. How much current it consumes dictates how much is available to charge the battery, which in turn dictates the size of battery you can support, which dictates the amount of current you can draw for your desired runtime, which in turn dictates the amount of charge current available, which determines the size of battery you can support, which dictates ... etc.  You get the idea.

You can support larger batteries, of course - I illustrated this at 0.2C charge rate (that's 20% charge per hour to give a 5 hour charge time). As long as the net charge over time increases (so during the periods between the 5-7 days of complete darkness you gain more charge than is expended during the dark period - something which has not been specified) then it should function fine.

Also note that your battery voltage should be below your solar panel voltage. A 6V solar cell would be able to charge a 4.5V lead acid battery (assuming low dropout schottky protection diodes). Also the capacity of the battery depends also on what you determine your "cut-off" voltage to be. That's how low the battery voltage is allowed to go before your system is determined to be unworkable. This could be at a point where the battery is in danger of being damaged by over-discharge, or at a point where your circuit no longer receives enough voltage to function (which could be linked to oscillator frequency on ATMega chips), etc. For this reason it is often better to choose a board that is designed to run at a lower voltage (3.3V) and use high efficiency switching regulation to get the most out of your battery. A 5V Arduino is seldom a great choice.

Finding that right balance is now entirely up to you.

And, in answer to your other question, yes there are ways to turn things on and off. You can "sleep" the Arduino. You can use high efficiency power circuits. You can add power switching circuits (P-MOSFETs) to switch power on and off to other modules. All things can (and mostly should) be done to save power when running on batteries.