What is this small strip of wood in front of fascia?
By location, the strip is called drip edge, but by profile, it is a 1X2. It should actually measure 3/4" by 1 1/2"

Cedar is a good choice to use, it is available already primed. If you choose to replace it, any cuts or exposed bare wood, no matter the location should be primed. It helps the longevity of the material.

This was not part of your question, but it is an observation of your overhang of the roofing material. It looks like the projection of the roofing past the 1X2 is marginal, in other words, it looks like it is not far enough out to protect the drip edge. I usually set my shingles at least 3/4" to 1" past the drip edge. This keeps the rain from getting the drip edge wet continually. I think that is what is making the material decay faster than usual. Fortunately it is not a structural piece, no more than supporting the lower edge of the shingles.



My first thought is, the roof projection without the drip edge will work fine, just touch up the fascia and be done. But if the drip edge is an aesthetic piece, as in covering up the exposed edge of the roof plywood, then perhaps a rake mold can work in its place. It is available in PVC plastic and will not rot, but it is very prone to extreme expansion and contraction, much more so than wood when exposed to the weather extremes.