Please help with insulation question

[intellectsucks]

I know absolutely NOTHING about properly insulating, but want to put some in this strange, 2'x2' little space in my kitchen ceiling against the exterior wall.  Some plumbing issues meant I needed to open it up but now that it is, I'd like to limit the amount of cold air getting in there.
Questions:

1. Which type of insulation will be most cost effective given a shorter time frame? (we don't plan on staying in the house for a long time)
2. How do I insulate it in a way that prevents moisture from accumulating?

Pics are attached and your feedback is VERY appreciated.

[ShoulderThingThatGoesUp]

To confirm, those stones we see in picture 2 are the structure and exterior of your house?

[zolotiyeruki]

Regardless of the materials used, there are a couple things you'll want to keep in mind:
1) keep all plumbing out of the insulation, and on the warm side of the insulation.
2) air sealing is very important.

Do masonry walls require a different approach for insulation, since they're permeable to moisture?

[intellectsucks]

To confirm, those stones we see in picture 2 are the structure and exterior of your house?

Yes they are.

[bobechs]

Stuff cats in there.  Cats are free and there's always more where they came from.

[Metric Mouse]

Ugg... the best way is to put a moisture barrier against the foundation to keep condensation from seeping into the interior space and saturating the insulation. Not easy to do unless you can crawl into that space the entire length. If you could get a barrier up, I would just blow it full of blow-in type insulation, since you only have a single access point. Any other option is going to be relatively time-consuming if you have to open up the whole space to do it "right".

[GuitarStv]

Ugg... the best way is to put a moisture barrier against the foundation to keep condensation from seeping into the interior space and saturating the insulation. Not easy to do unless you can crawl into that space the entire length. If you could get a barrier up, I would just blow it full of blow-in type insulation, since you only have a single access point. Any other option is going to be relatively time-consuming if you have to open up the whole space to do it "right".

+1

There's no really good cheap/easy way to treat this space.  Barrier + fluffy is an OK short term solution.  Don't use the fluffy stuff without a moisture barrier though, because if it starts to wick moisture from the wall it could cause all sorts of problems.

[Ottawa]

I would consider closed cell spray foam insulation.  This will provide a moisture barrier and insulation. 

[RichMoose]

I would probably go with Roxul. It's reasonable, easy to install, and performs better than fiberglass even when it's damp. Putting up a 7m poly barrier would be a smart move.

[lthenderson]

I would consider closed cell spray foam insulation.  This will provide a moisture barrier and insulation.

Probably a cheaper and easier install would to just buy some rigid closed cell foam in a sheet and cut pieces big enough to stick up in your hole and maneuver in place against the outside wall with a pole of some sort. If you can get it in place fairly easy this way, I might apply some construction adhesive to the side that will butt up against the wall to keep it in place.

[Noahjoe]

the only real way you can do anything without cutting more holes is to blow it in place. I'd do fiberglass before I did cellulose because cellulose is gross, dusty, settles, and will probably wick more moisture. It doesn't look like the 2x4s would support a person crawling around up there.  All of the suggestions are good, but there's no way you can do anything with foam (sheets or spray) or really even a moisture barrier for that whole length without opening it up. If you're not going to open it up, blow in fiberglass and call it a day. It'll be worlds better than it is now.

If you want to do it right, I'd open it up and probably do the foam sheet against the wall. even 1" (R6 if you do blue/pinkboard) would be a huge step in air sealing it. Then I'd Great Stuff around it (Great Stuff is actually closed cell). From there I'd tack in a barrier on the lower level to keep it from falling out if ever you are poking around in there again later. Then i'd blow in a ton of Fiberglass to desired R-value. Or you could foam board it with 2x2" foam board for an R-20 value, though that stuff gets expensive fast. it'd be easy to install though if you opened up the space a bit more. I'd still Great Stuff/air seal around it.

My .02. It'll cost you a little more to do it, but it'll be done right. I get so, so, so mad when people shortchange themselves (or the next homeowner) on projects like this. I used to sell building materials, so it's a sore subject for me ;)