It's an easy fix. However, note that the internals can retain dangerous voltages even when the TV is off. They could maybe even kill you. Make sure you understand what you're getting yourself into. Read about the risks and how to stay safe, intelligently decide whether to do this yourself, and be very careful.
The back is pretty easy to remove. Many screws around the edge connect it to the plastic front, and three screws above the back connector panel connect it to the main board assembly. The TV can stand on just the plastic front, so the back can be removed while it is upright. I kept it near the edge of a workbench. This allowed access to the screw at the bottom, and when the back was removed it allowed access to the bottom of the main board. That position seems precarious, but the TV is not too close to tipping over. Make sure that the table is not close to tipping over. Also, when removing the back, make sure that it does not fall and strike the board at the back of the picture tube neck. If you need a service manual, the 27AX5BX manual will probably suffice.
Here is a the main board. It is on rails, and held to the front plastic with two screws at either side. If you unscrew those, you can slide the board out a bit for better access to the bottom. It's worth doing that, because working on the bottom is much easier then. You could even unplug cables and take the board out, but it's not necessary for access to the vertical deflection circuit.
C627 is the pump-up capacitor for the LA7838 vertical deflection chip, and it is the main suspect. It is part of a charge pump. The chip first connects the capacitor to the power supply to charge it, and then it connects the capacitor in series with the power supply, to provide twice the power supply voltage. This higher voltage is used to quickly move the beam back to the top. A problem with the capacitor will mean less power is available for moving the beam back to the top. This will slow down the move, and the beam won't be able to reach the top before it's time to start displaying the next frame. C625 The small capacitor with the brown ring is a power supply filtering capacitor. It could contribute to the problem if the power supply dips when the vertical deflection chip uses a lot of current.
When I looked under the board, I saw a surprise. The trace between the pump-up capacitor and the chip was missing, and one of the capacitor leads was used to make the connection. This plus the different look of C627 made me wonder if this was a previous repair. Could the pump-up capacitor be good, and could the puddle-like mark on the board be from a failed previous pump-up capacitor?
tested the ESR of the pump-up capacitor. Although the capacitor still had most of its capacity, the ESR was ridiculously high, and so the capacitor was obviously bad. There was no need to test the small capacitor, because it was leaking badly. After I removed it, there were drops of electrolyte on it.
After cleaning the board and replacing C625 and C627, the TV worked properly. I chose to not replace more capacitors. Yes, it's probable that some other ones are degraded, but there are very many and it's not worth the effort. The real problem here is that a lot of electrolytic capacitors are located very close to heat sinks. Hitachi even used 85°C capacitors there!