With 397 rolls of a fair 6-sided die (not my favorite use of time, but certainly not infeasible), I could generate more than 1024 random bits. Could I use those bits to generate a keypair that could then be used in encryption without ever touching a (p)rng again?
There are two points:
Generally speaking, cryptographic life without randomness is harsh. A lot of algorithms and protocols require some randomness (e.g. symmetric encryption in CBC mode should have a random IV). You can have "sufficiently good" randomness in a given system as long as you have a secret key, and either state or a non-repeating public value (e.g. a clock, but beware of resets). Indeed, if you have a counter, then you can encrypt the successive counter values with a block cipher (AES comes to mind) and this will be a good PRNG; a non-repeating value known to everybody (such as the current time) can serve as a substitute for the counter.
If a system has no state to update, and no source of randomness, then it is intrinsically rewindable: boot it up, then send the same inputs as previously, and you will get the same output. This is what it means, when we say that a computer is a deterministic system. Replay attacks are the main worry here.
Assuming your dice is fair, each dice roll yields around 2.585 bits of entropy. You could use this to generate strong key-material.
Generating asymmetric keys is however a mathematical process. Both RSA and DSA key generation involve selecting primes. So you can't approach the process with a huge random number in hand and expect to be able to use it as-is.
There are opportunities to 'mix in' your high-entropy dice-material during the key generation process, namely when selecting your primes and encryption exponent in RSA, or when selecting the primes or private secret in DSA.
You would not need to generate so much however, a 1024 bit RSA key holds around 80 bits of entropy;
So you would only need 80 bits of entropy to generate a 1024 bit RSA key. That is to say, that if you consumed as much randomness as possible during every stage of key generation, you would find that you had used up around 80 bits of dice-material by the time you had finished generation.