Given a private RSA key, how do we get the public key? Is it possible to pre-choose a private RSA key, then obtain a public key from it?
Yes, in general the private key is generated at the same time as the public key in RSA, first the $p$, $q$ primes, then $n = pq$, then the public exponent $e$ and finally the private exponent $d$.

If the private key is given as $(p, q, d$) then you can recover the public key easily. As long as you know the public exponent and $n$, you have the public key (and this means that if you cannot guess $e$, you need $d$ and the factorization of $n$, this is an instance of the RSA problem).

However, as @fgrieu points out, if the private key is stored as $(n, d)$ and $e$ is not conventionally chosen and unknown, then the public key is lost and cannot be obtained from the private key.

The opposite (private key from public key) is meant to be infeasible though.