Hello Panos and welcome.
You are right, the bus doesn't go to Nimporio so your only option is a taxi or you could walk (it isn't too far). I've never used a taxi on that route but judging by other journeys on Symi I'd expect it to cost 4-5 Euros but that is a guess, other poster her may have more accurate information.
All the taxis have meters so you won't be over-charged although they may prefer to fix the price with you before you set off. Just ask the driver.
I should warn you though that taxis can't get right down to the very bottom of Nimporio because the road isn't suitable. It's unlikely that your accommodation will be that far away but if it is you may have to walk to meet the taxi.
You may be in luck with the ferry situation. The Sea Dreams
Symi ferry is scheduled to leave Akandia port in Rhodes Town arriving on Symi at 21.00 on Sundays up until September 10th. Have a look at the updated ferry schedule on our 'Noticebord'
http://symiforum.com/viewtopic.php?f=8&t=2863