We have only a half day to spend in Venice, from noon to 9pm. How shall we spend it?

Just wander the city and eat wherever you're hungry. You might want to read Rick Steves' guide around Venice. I think you can see a lot in your small time. His guide will give you the highlights.

Half a day. That’s tragically short!

However,
Take Vaporetto No 1 down the Grand Canal from the station (assuming you arrive by rail) to San Marco. You’ll pass under the Rialto Bridge.
Visit the Basilica – but skip it if the queue is too long, and return back there later in the afternoon, say at 5:00 when the crowds start to thin out.

Admire (or quietly ridicule) the cafe orchestras at Florians and Quadri’s in the Piazza San Marco. Maybe have a coffee in the Piazza at an outdoor table. This will not be cheap – think 30 Euro for two, but it is quite an experience.

Admire the facade of the Doge’s Palace, walk along the Grand Canal away from San Marco, see the Bridge of Sighs.

Walk back into the Piazza, leave via the entry furthest away from the Basillica, towards Salizada San Moise. There’s a magic glass shop on your right after a couple of hundred yards. Not cheap, but pretty good quality – modern art museum quality.

Keep going towards Campo San Stefano, and Academe. Cross the Academe bridge, which gives a good view of the grand Canal and the lagoon. The Academe gallery is in front of you if you are interested in art. The painting “The Feast at the House of Levi” is special.

From there, you can find your way to the station – follow the signs saying “Ferrovia”. Have a spritz in any of the bars in Campo Margerita, maybe a meal there.

I’d skip the Doges Palace – it really needs a couple of hours to even start to see it. The Rialto Bridge and surrounds can be very crowded during the day – you don’t want to stand in line to cross the bridge.

Brilliant!

Assuming you will be coming soon (before October?), with the longer Summer days, you might consider organizing so you are finishing the day towards the Zattere area of Dorsoduro, with a meal overlooking the Giudecca canal and the setting sun off the Lagoon, and then finishing it off with a gelato at Nico's (the Gianduiotto da Passeggio, basically a "half-brick" of hazelnut chocolate drowning in fresh whipped cream, is "to die for") before catching a vaporetto back to your departure.
Just a thought