Code P0301, mileage, 12V battery. Related?

Hi all,

2008 Prius standard (lowest trim), 150K miles.

Not terribly mechanically inclined here, but I like to make sure I have an idea of what I'm talking about before I take it into the mechanic.

My car first threw a P0301 code about 6 months ago. I was due for an oil change then, and did a little research on the code. Spark plugs came up as the #1 cause and I knew I was about due to have them replaced anyway so I did that at the same time, and all was good for three months. The code had cleared itself before I got into the mechanic so I didn't worry too much about it.

About 3 months ago, it threw the code again, and again cleared itself. However, it has now happened a few more times, with increasing frequency. Almost without exception, it happens shortly after startup.

There's no apparent change in behavior; idle is subjectively a little rough but it always has been so I don't think too much about it. I've had the car since 45K, the only unplanned mechanical work I've needed done was a coil replacement around 50K.

In parallel, my mileage has dropped fairly significantly, from about 45mpg on a tank to 38mpg. I can attribute some of this to moving closer to work and having a shorter drive, but I don't feel like THAT much, it's still a 15min/10 mile trip with some freeway miles. I've started looking into that, and doing the on-screen battery test shows 11.7V after an overnight, 11.4V after hitting the button again to turn everything on. The 12V battery is about 8 years old at this point and has been drained at least once since then, so I'm guessing it's about due. Agree?

So now for the real question... what is the chance that all of these are related? I feel like it's pretty low since I don't believe the 12V inputs to the coils, but I'm not 100% sure there. Also given that I'm ONLY seeing the code for cylinder 1, I feel like it must be something specific to that cylinder. Anything I should ask the mechanic to look at specifically, or red flags that they're trying to jerk me around?

Thanks!