Hi there,
Couple of things to note here.
First, we dont know the inductance of your inductor. This is a problem because we would like to be able to calculate the ripple voltage, or at least estimate it. We want to do this because there is an issue that comes up when charging batteries like these with a switching power supply type charger, and that is that the ripple voltage should not push the output voltage above the 4.250 voltage limit of the cell. Really though, 4.200v is a better limit that should be imposed (assuming your cell is a 4.200v max type cell).
Second, that chip you selected unfortunately produces higher output ripple than most regulator chips. The reason for this is because it is based on a digital regulating principle rather than the more conventional analog method. The digital regulation is much different than analog, and the byproduct of this is more output ripple for the same filter component values. The only way around this is to make the inductor larger than usual. For example, maybe 1000uH rather than 100uH might accomplish this, but the final design needs to be tested to make sure it does not take the cell voltage above 4.200 volts at any time. Unfortunately we can not calculate the inductance of your inductor without knowing what kind of material the core is made of, so you are better off testing the design rather than trying to estimate the inductance, unless of course you want to try to measure it but an inductance meter probably wont show the true inductance for this kind of inductor so you should use a scope instead.
Third, if you are going to charge at 1 amp with a 4v output and you have a 16v input then theoretically you only need one quarter of the input current. That's because a buck converter is a true power converter so if you put in 4 watts you can get 4 watts out even if it is at a different voltage. Because of inefficiencies however, you'll need a little more input power than output power, but if the design is set up right you can easily get away with a 16v at 0.5 amp dc wall wart and you should easily get 4.2v at 1 amp output. This of course means you dont need such a heavy duty wall wart rated for 4 amps. This also means we get a little bit of 'hard wired' current protection for free.