$\mathrm{p}K_\mathrm{a}$ of methane is $\sim 50$ and that of ammonia is $\sim 37,$ thus ammonia is a better acid than methane, which implies methane is a better base than ammonia.
But I don't see how: methane doesn't have any lone pair of electron, neither it has any $\ce{OH-}.$ How is it possible?
You got it wrong. Acidic and basic strength of a compound are unrelated, about as much so as the man's name and his weight. Within the definition you and I are currently using (Brønsted–Lowry), acidity is about having a proton on a polar bond ready to ionize, while basicity is about having electron pairs ready for accepting a proton. A compound can have one and not the other, or vice versa, or both, or neither. Why should the two be related? They shouldn't and aren't. Well, there is one obvious exception: a strong acid can't be a strong base at the same time, otherwise it would turn upon itself. Except that, everything is possible.
Indeed, methane is a very weak acid, about 13 orders of magnitude weaker than ammonia (as attested by the difference in their pKa). In fact, it is one of the weakest acids among all compounds. This is not unexpected, given its almost non-polar bonds and the lack of stabilization in the anion. Also, methane is a very weak base. Forcing it to accept a proton was no small achievement in itself (see Methanium). This is not unexpected either, given its lack of lone pairs or negative charge.
Now you must be thinking that I am pulling your leg, because every kid knows that strong acid means weak base and vice versa. It is written in every textbook, said by every teacher, and that with great confidence, too. Can they all be wrong? No, they are right. How so?
Here's the trick: strong acid does indeed mean weak base, but not in the same compound. Look again at that equilibrium: $$\ce{HA <=> H+ + A-}$$
See that? It is $\ce{HA}$ that acts as an acid here. And now look at the reverse reaction: it is $\ce{A-}$ acting as a base. Now if the equilibrium is shifted to the right, then $\ce{HA}$ is a strong acid, which inevitably means that its conjugate base $\ce{A-}$ is a weak base.
So it goes.
When A is considered a stronger acid than B, it does not follow that B is a stronger base than A. Rather, the appropriate base comparison is between the conjugate bases formed from acids A and B. Properly, the conjugate base of B is a stronger base than the conjugate base of A.
In terms of the compounds given in the question, the conjugate base of ammonia is amide ion ($\ce{NH3<=>H^+ + \color{blue}{NH2^-}}$) and the conjugate base of methane is methide ion ($\ce{CH4<=>H^+ + \color{blue}{CH3^-}}$). Therefore, if ammonia is a stronger acid than methane then methide ion is a stronger base than amide ion, regardless of how basic the initial compounds methane and ammonia might be. Which is true, as anyone knows who has tried to combine a Grignard reagent with a compound having hydrogen bonded to nitrogen.
