How to analyze a capacitor connected to a DC voltage directly (no resistor)?
Good question. If you connect an IDEAL voltage source to an IDEAL capacitor you get exactly the situation that you describe. The IDEAL answer is that there will be an INFINITE amount of current for an INFINITELY short amount of time to get it all sorted out.
In practice that doesn't happen since IDEAL voltage sources and capacitors do not exist. All real voltage sources have some finite (non-zero) source impedance which limits the current and results in normal exponential charging of the capacitor. The smaller the source impedance the quicker the capacitor will charge and the higher the initial charging current will be.
And there is always option B: something is just going to blow up. Connecting a large uncharged capacitor to a beefy power source with a very low source impedance is best observed from a safe distance until the fireworks have subsided. In other words: DON'T DO THIS.