A tarp that fully covers a bike can provide some insulation and keep out precipitation. Don't buy a "bike cover"; they're overpriced, thin, and non-customizable. For an average-sized bike, an 8' by 10' tarp is enough to just reach to the ground. A weight of around 6 oz per square yard, or a thickness of around 8 mils, seems to be ideal (standard tarps are around 3 oz per square yard and 5 mils thick). In theory, silver colored tarps reflect more infrared than other colors and are better for insulation, but I'm not sure how much of a difference this makes in practice.
One of the most important things to look out for in a tarp is that the grommets are spaced no more than 18 inches apart. You can bunch up all of the grommets along each short end of the tarp and use cable ties to semi-permanently join these grommets. This gives the tarp a fitted shape and makes it fit nicely around the wheels. A good starting point is joining all of the grommets along each short end to a single point. Once you've tried this, you can experiment with tying grommets together in various configurations with cable ties to see if you get a better fit for your bike. You can also tie grommets to a bike rack, which can make it easier to get the bike in and out if you're riding it daily.
The following picture shows a fitted tarp on a bike. On the near side, the four middle grommets are tied together at a point, and the remaining two grommets (at the corners) are tied at a point just below. On the far side, all six grommets are tied together at a point and tied to the bike rack. The long sides of the tarp in the picture are folded in a bit; without folding, they just reach to the ground. With grommets spaced 18 inches apart, you have flexibility to customize the fitting.
In summary, if you get a tarp, get one with the following specs:
- 8 by 10 feet
- at least 6 oz per square yard (200 gsm) or 8 mils thick
- grommets spaced no more than 18 inches
- silver color (?)
and then tie grommets together with cable ties to give it a fitted shape.
As I've been using this for a while, I've noticed that air flow doesn't seem to be much of a problem in practice, so I usually let the sides of the tarp reach to the ground. The tarp is stiff enough that there are always a few small openings where air can get in and out but water can't.