The circuit that you referred to is not going to work very well, regardless of whether you use a bipoalr transistor or a MOSFET. That's because you are trying to do High-Side control with a NPN or N-channel device.
Because you are working with a solar panel, you have two options: shunt regulator or series regulator.
A shunt regulator makes use of one property of solar panels: they function somewhat like a current source. That is: for a given amount of insolation (the amount of sunlight hitting the panel), the current remains roughly the same as the terminal voltage varies. A solar panel can usually run with a direct short on its' output leads without any damage.
The advantage of a shunt regulator is that the negative lead of the panel can be connected to your circuit ground and still allow the use of a NPN transistor or N-channel MOSFET to provide the short across the panel. Obviously, there is a series diode from the junction of solar panel (+) / transistor to the battery. This diode is needed anyway so that the solar panel doesn't discharge the battery when light levels are low.
Because the shunt regulator has to dissipate all the unwanted power as heat, the most common shunt regulator configuration is the "bang-bang" controller. This is where the shunt is either fully OFF (allowing maximum possible charge current) or fully ON (solar panel is shorted, resulting in NO charge current). This results in minimum heat in the switching device. Many inexpensive solar charge controllers work this way.
The other option is a series regulator. Now you have to make a choice: you can use NPN bipolar transistors or N-channel MOSFETs as the pass element BUT you have to control the negative lead of the solar panel. In other words, the positive lead of the solar panel connects directly to the battery (+) terminal (through a series diode if necessary). The negative lead of the solar panel connects to the drain of the N-channel MOSFET, with the MOSFET's source terminal going to circuit ground.
I mention that the series diode on the (+) lead of the solar panel might be optional. That's because it may not be needed because you can turn the transistor / MOSFET off when charging isn't possible because of not enough light on the panel.
If you do want to go with a N-channel MOSFET being controlled by a microcontroller, my "go-to" part for low-voltage, medium current DC switching is the IRF3708. 30V, 62A continuous, 0.012 Ohms Rds on. Drive the gate with a 47 Ohm resistor mounted as close to the gate as possible.