The areas will be covered by deep eaves, but still outdoors.

What type of finish (if any) should I use? I want it to look old, but at the same time, I don't want to replace it in 2 yrs either. What type of shrinkage should I expect? This will be uncured, rough sawn lumber.

Thanks for your advice

Red oak doesn't handle being wet very well. If these posts tend to get wet often, white oak would be the better choice.

Shrinkage depends on the moisture content at installation and, to a lesser degree, on whether these posts get much direct sun or are always shaded. "Uncured" isn't nearly specific enough to make predictions. If the wood is freshly cut and soaking wet, shrinkage will be considerable. If it's been stacked outdoors for several months, shrinkage may be negligible.

The boards will be freshly cut, though I don't know how long the logs have been sitting at the mill. There will be very little direct sun exposure. Should I stack the lumber on the porch for a few weeks prior to installation?

A few weeks of sitting on the porch is not enough to save you from this issue. Fresh sawn, the wood will be sopping wet. Half of its weight might be water. Much of that water will evaporate in those few weeks on the porch, but the shrinkage won't even start until the moisture content gets below 30%, by which point drying will slow considerably. FWIW, a common rule of thumb is that air drying lumber takes about a year per inch of thickness.

In short, I wouldn't do what you're thinking of doing.