I got this little pepper plant recently in now the peppers and some leaves are withering, should I cut those part off?
Yes, take the withering fruits and leaves as they should come out very easy. Also, take all the leaves that are on the soil because they can be used as incubator by microorganisms.
I notice the leaves have holes, probably because of insects. If you don't want to use pesticides (if you do, keep in mind that you have to wait a few weeks before the fruits should be harvested), you can clean your plant leaf by leaf with a damp soft cloth. Your plant needs to do photosynthesis and it can't do that if the leaves are covered with dust or pesticide white drops.
If the white spots on the leaves are not insects, there may be a fungus and in this case don't use a damp cloth for cleaning because this will spread the fungus everywhere.
First, that pot is too big for your plant. It should be half that size. You do have white fly and they suck. Seriously. Get NEEM and make a solution 1 gallon to one ounce then tip your plant completely upside down and immerse in the solution. Swish around. If you need to, get a paper plate, cut a slice to the center and fit around your plant to cover the soil.
Harvest your peppers first. You've got one already in the drying phase, all of those peppers should be cut off. That will allow the plant to put energy into insect repelling. Douche the top of the soil shallowly with the neem mixture and get rid of the rest of the mixture by dumping in a hole somewhere in your yard. You don't want to kill other organisms or at least kill as few as possible. Neem is fairly ORGANIC and is as strong as one needs to use.
Leave the plant in its pot IF YOU HAVE USED POTTING SOIL. If not, and you have used garden soil, get that plant transplanted into a pot a bit smaller (4 or 6 inch diameter) using sterilized potting soil! No rock or gravel below the soil and above the drainage hole. Get the bottom of the pot off the surface to enhance drainage.
Baking soda does nothing. When the solution dries you'll get a white coating. No big deal except baking soda will raise the pH of your soil and peppers like a bit acidic of a soil (6.5). Remove those peppers and use them, your plant might just produce more that way. Is this indoors? How did you get your flowers to be pollinated if so? What kind of light are you using? If you are indoors you need to get a decent grow light. Daylight hours are short here in the northern hemisphere and you won't be able to produce flowers and fruit without a grow light. Are you in the southern hemisphere? You'll have to manually pollinate which is not a big deal. Let us know.
Cut lower leaves off, but leave the others. If you don't have decent photosynthetic growth forget getting any more flowers, fruits, peppers. If you do not have decent light and long enough daylight hours forget anymore production as well. How long have you had this plant indoors? You've gotten a nice little crop of peppers. More than most people trying to do this indoors!! Where do you live?