I think the cabinet looks fine as is. The backsplash is the main thing I see that could be updated. If redo backsplash, keep it under cabinets. Don't need to come out on adjacent walls unless water near by.

Maybe there is also a more modern way to the do the back splash too, but I definitely want to update the honey oak finish cabinets.

I would also change the cathedral arch upper doors, but I think that may be getting into more work and cost than want. So, that leaves just painting them.

I want to put less than $5K into the kitchen and it will not be DIY.

Do you need the uppers on the window wall? By removing them, adding a few floating shelves and changing the back splash, I think your kitchen would look spectacular.

There will be no DIY involved in this.

I recently had the walls repainted and I had the painter paint the honey oak linen closet doors as a test to see if I liked the way they came out.

I liked it, so now I want the kitchen cabinets also repainted.

So your entire budget will likely go toward painting the cabinets. I agree that the backsplash could use a refresh as well. Start getting quotes from pro cabinet refinishers - not house painters - and go from there. Follow artistsharonva's advice on running far from anyone who doesn't specify all the steps that will go into the prep and finishing. Do some searching on here to find out what to look for and what to avoid as there are many stories which detail problems from bad paint jobs.

This is how the linen closet doors came out.

So, what are better back splash ideas that would fit this type of kitchen?

If I get new back splash that doesn't cover all the existing areas, it will need new drywall, restexturizing and painting,

I have seen several threads re: painting or whitewashing backsplashes. Something to consider having your painter do.

Painting over the back splash to make it match the wall color would be an improvement?

Painting over the backsplash won't update the look. If you hate the oak, go ahead with the paint, but paint all of the cabinets....just know that it may not look as updated as you'd like because of the design of your cabinet faces. I also fear that if you use the color of the shutters for the cabinets, your floor and backsplash may look "dirty" so be sure to check lots of whites and compare them to all of the elements in your room.

I have seen painted cabinets with the same arch design and still felt it was an improvement over honey oak.

I'd get opinions on whether getting charcoal or dark gray or charcoal on the lower cabinets would look better than getting them all painted the same white color.

What about painting cabinets to match the wall color instead of white?

Get knobs for sure, that will give it a huge face lift!
you could always paint that tile backsplash as well.

I think if you paint the uppers you’ve got to paint the lowers. Different uppers and lowers are fine, but stain on one and paint on the other are hard to pull off, especially if the stained lowers are not really dark. Know that you may have cracks in future where the painted panels meet the rails and stiles—panels are meant to (almost microscopically) expand and contract, and it shows on painted finishes. And I think the backsplash is fine. It’s not my liking (but it’s not my house so who cares), but its hardly dated and really, are we to rip backsplashes out every 5 years as trends change? PS might there be a better home for that fire extinguisher?

I was always going to paint all the cabinets. The only question was which colors and whether to do all one color.

Unless you have cabinet people repaint the cabinets and not a house painter, your cabinets will look like garbage in a year or two. There is a whole process to painting cabinets correctly and that process will run you between $3000-$5000 for your small kitchen.

The house painter painted 2 linen closet doors and they look fine, but it has been only 2 months.

What specifically will a cabinet painter do that a house painter will not or cannot do?

I think white uppers and charcoal lowers would work nicely in your kitchen. Your cabinets look like builders’ grade so I would paint them.

I'm going to be contrarian here and say - I had my kitchen cabinets painted when we first moved into our house, and it was fine. I believe the painter used alkyd paint; we did white on uppers and lowers (because our kitchen was super dark) and it held up for a good four years until we renovated the whole thing.

Adding modern hardware on the doors and accessorizing the top of the cabinets would make a big difference already.