This is a really clever idea that's theoretically possible as the sun would indeed destroy anything sent toward it. Once in space, a ship would initially orbit the sun along with the earth, but it would be relatively straightforward to slow the garabage's orbit and launch it into the sun.

Though I should note that compressed air, while convenient for household applications, wouldn't be a very effective propellant for this kind of thing-- thankfully, modern rocket fuels are significantly more powerful.

Probably the largest obstacle to something like this ever happening (at least in the imaginable future) is that it's extremely expensive to send things to space, requiring enormous amounts of energy to escape Earth's gravity. Currently it costs about $10,000 per pound just to get into orbit! While landfills are terrible and environmentally damaging, making the energy to send their contents into space would be even more disastrous.

Another concern is space debris, which are already a serious problem from the small number of satellites and other ships that have blown up in space. If a garbage spaceship exploded in Earth's orbit, its cargo could get stuck floating there, which would be dangerous for future spacecraft and satellites.