There really isn't much to do

. You can take a walk and see the whole city in an hour, two hours max.
Many companies base procurement centers in Luxembourg to avoid VAT -- hence, a lot of people are there during the work week and commute home on weekends. Things tend to close early and to be quiet on weekends. Public transit is free on weekends to encourage people to visit

.
Are you doing Luxembourg as a stop-over in between other destinations? If so, I would recommend planning it as a lunchbreak. Take a train or bus that arrives around noon and leaves around 4 -- you'll have plenty of time for a leisurely lunch and to see everything. Or if you're staying in Paris or London, take a train or flight that gets in early, and then you'll be ready to leave by lunchtime.
If you find yourself with extra time on your hands, you can take a bus to Schengen, where the famous agreements removing Europe's internal borders were signed. There isn't much there either - mainly a small museum. You can walk across the bridge to Germany, and then be in France five minutes later (on foot).
I did all of the above as a daytrip from London -- woke up and slept in my own bed -- and had time to spare

. Don't get me wrong, it's pretty -- there's just not a lot to do or see.