We are 3 persons travelling on budget tour in Europe in November, and will be staying in Cologne from 10th to 12th Nov. evening. I very keen to visit Amsterdam. But I have only one night and one full day in hand. Can any one give idea on how quickly can I go to Amsterdam and Brussels in one day esp. on 12th November. Of course I will choose the least expensive option.

I think you are better off enjoying Cologne tbh. The train from Cologne to Amsterdam takes approximately 2hours 40 minutes, and unless you can book it well in advance is not that cheap. It is the fastest way to travel between the two cities.
To then add Brussels into the mix adds more train travel, and more expense. Plus you will presumably need to find a place to stay in Amsterdam or Brussels, neither of which is an especially cheap city, though prices have fallen somewhat lately.

Have a look at www.nshispeed.nl/en or www.bahn.de for trains, times and prices so you can decide for yourself if it is really worth it.

While Brussels is nice, the idea to see Amsterdam AND Brussels in one day is ridiculous. All you see will be the surrounding of the train station.

Go to Amsterdam for a full day - it is still just scratching the surface.

You cannot get to both these cities in the time you have, they are roughly equidistant from Cologne, but in different directions. Going to just one may be a bit crazy, but that's for you to decide (it takes one to know one...)

You can get from Cologne to Bruxelles on Friday Nov. 12th in the evening (that's how I understand your post - am I right?) for 19 Euros with the ICE that leaves Cologne at 19:43, you get into Brux. Nord at 21:27, switch to the local train that leaves for the center (Brux.-Central) at 21:32, and you're within walking distance from the fabulous Grand' Place at 21:35!

You want to book a cheap hotel in the vicinity of the Grand' Place, we stayed in the Hotel Mozart one time, but I think they were renovating it and it could be much more pricey these days.

The Grand' Place at midnight is just awesome, you feel like transported back by a few centuries. All around it are tons of restaurants of all price classes, and hotels and what not - it will be very lively until very late on a Friday, like in the Rue du Marché aux Fromages 15 (which means Cheese Market Street...it's a pedestrian-only street these days) and thereabouts. If it gets really cold people will go inside, but the outdoor cafes are usually packed.

You can get back for 19 Euros on Saturday Nov. 13th by leaving on the 18:10 from Brux. Central to Brux. Nord again, then pick up the ICE that leaves at 18:33 and gets to Cologne at 20:15.

I found these at www.b-europe.com/Travel - go for it, it looks like you can book these online.

Check out the Amsterdam trip on the same site - I found trains but it says that you cannot book that route online, so you'd have to probably pay a lot more if you buy the tickets at short notice.

You can also look at www.bahn.de - the German train site, maybe they will let you book online.

Good luck!