I own a Classic 350 which I have ridden only 20,000 km in the last 10 years. I want to buy a faster bike, like the Duke 390. However, that bike will also be used only once a week. Should I buy a new one, or is a pre-owned bike a better option?

A new bike will come with a warranty, and you will be the first one to break it in, so the peace of mind of buying a new machine is not to be discounted. 

If you are looking at older bikes that are 4-5 years old or more, then you will also need to budget for a comprehensive service and perhaps some replacement cycle parts like tyres and brakes. 

While the KTM 390 Duke is a great option, you could also look at the Royal Enfield Guerrilla 450 since you've already owned a bike from the same brand.