Q: No power to cassette and dash time

I bought an 89 Honda Accord a few months back and the original owner told me that the cassette player no longer worked so I figured the player was dead. Well a month or so ago I started having starting problems and needed a jump from a buddy to get it home. On the drive home though I noticed that the cassette and dash time appeared to be working for the first time since I've had the car and they were working properly. But as soon as I pulled into my drive and sat idle for a minute or so they both dimmed and faded as if they once again no longer had power running to them. This was the only time I have seen the player have any power to it. I traced my starting problem to a bad battery and have had no problems starting the car since but I still have absolutely no power to the player or the time on my dash. I've checked the fuses of which appear to be fine and the alternator which also is running good. I am lost as to what could be causing this, any advice is appreciated thanks

My car has 150000 miles.
My car has an automatic transmission.