Training for 340km in Greece I'm flying to Athens, in 5 weeks. I'm going to do a 340km route from Athens to Patras and back, on the both sides of Ionian Sea. 

Right now my max achievable distance is 200km (trekking bike, avg 24km/h). Two weeks ago I began doing gym twice a week, spinning twice a week, and long (100-200km) ride during weekends.

Are those 5 weeks enough to train myself for such distance? (with decent avg speed)

Also, is this route a good choice for late October? I dont want huge cold wind blowing on my face the whole day.
Try it out! For long distances it helps me personally to remind myself that anybody can cycle 100 km when I only have 100 km or so left. It may hurt, but really, any cyclist could do that. This turns your ride into 240 km, which is only 40 km more than what you seem to be comfortable with. And of course anybody can do 40 km ;-)

It all happens in your head, especially when you leave your comfort zone...

Planning/scheduling also helps me a lot when covering long distances more-or-less non-stop. Make a conservative schedule, e.g. based on 21 km/h (or whatever is "quite slow" for you) and taking planned longer rests (one or two?) into account. Based on that figure out your scheduled time for each 20 km and write it somewhere accessible while riding. If you planned conservatively, you will be able to stay ahead of your schedule. Then you can stay calm, and just ride with the feeling that you will manage. Use some of the "buffer" for smaller breaks.