you should watch a few videos on Utube. Look for Faux wood graining. You can get the rubber tools from any good wood finishing supply place and then just find some scraps to practice on.
as an example
https://www.youtube.com/watch?v=l3JHAGo9-m0It sounds like MDF so you can't strip it. If you have a tablesaw you could also cut your choice of wood into thin "veneer" maybe like about 1/8" thick. Just glue it to the top and front, then run a roundover bit over it. if it's just a simple molding. I've done that and made up a hollow box for some of my trim over the windows as it needs to be 1" thick and I found a board I liked but it wasn't enough to make three so I sliced it into three. If you match the grain pattern and use all wood from the smae plank you can do it so no one notices it's not a solid board.
what I do is put he board over the tablesaw and make two cuts one on each side to slice off what I want , then use a handsaw to finish the cut if your blade won't lift high enough. I have a little thickness planer and if you have one there is a trick. Take a piece of flat MDF then glue the 1/4" thick piece that you cut off with hot melt glue. You only need a couple blobs to hold it from moving. then you can run both boards through together and that will make one flat side on the thin board that you cut. then you break the glue and you can flip the thin board over and make the two sides flat and even thickness to each other. If running thin stuff through the thickness planer is scary then you can glue it down a second time so that again both boards go through the planer together. the little rigid or dewalt or makita lunchbox planers are fine for the home workshop. In a commercial shop they may have better planers and sanders to do this but you can do it at home for not too much of you have the room and use for a few tools.
You can also get veneers and they have iron on ones or you can glue them on. since modern veneer is thin you'd need to glue a strip of real wood that matches to the top edge so you'd have enough "meat" to cut the roundover. the back and bottom side don't need to be veneered.
when veneering tables and things it's common to veneer both sides so it is even and lass likely to warp but I think I'd just paint the back. it gets nailed to the wall and I don't' think you'd have issues with warping. If it were a table or a shelf or something I'd veneer both sides.
or just hunt around. check craigslist or if you have an old house parts place they do get pulled out of houses they knock down. Ive replaced a few that way.
This summer I was lucky to get all the trim from a craftsman house. I didn't go in the house but I guess they just modernized it. Ive done that for most of my missing ones. when I did my attic I wanted it fast so I went and bought 3/4" edge grain knot free fir. the stuff is expensive. If I did it again I'd buy the boards a tad wide and then use that piece I ripped off to glue behind the top edge so it appeared to be 1 inch,, but i am happy with it since it was nice wood it doesn't look really out of place. It is expensive wood though..
there is noting wrong with the inside parts being MDF if you just cover it with real wood you;d never know unless you get a flood but its unlikely you would have an issue. If you didn't like using MDF you could just use pine or whatever you get cheap.
another way is to see if you have a small local mill around. they will cut custom orders and may even stock 5/4 (or 1 1/4") rough sawn lumber. I did that because I wanted 1" boards for the risers on my stairs. I got the wood green so I had to sticker and stack it and put a fan on it for a month or two but if you have time for it to dry, great. it did require trimming the edges on the tablesaw and I ran it through the thickness planer to get smooth concentric boards. By buying it rough sawn I didnt' pay so much and if you do that you cna get some extra for other projests and so you can high grade the stack a bit. what I do is order more and then select what I want and pieces with bad knots and things I trim down. large mills have lumber graders that separate the best boards out and charge high prices for knot free edge grain stuff. If you deal with a small mill youll get a more mixed run some boards may be unusable and others may be "perfect" I save the perfect ones for things like window sills and stuff that shows. this way I get some wood I really couldn't afford to buy otherwise. Of course the quality depends on the logs the mill is able to source but some cater to this sort of thing and it can be way cheaper than the lumberyard. forget the big box places.
anywhere there are a few ideas to explore if you choose.
Phil
