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I've read about the "Weak Diffie-Hellman" attack (paper, website), where a resourceful entity like a state can pre-compute values for known primes to aid solving the discrete logarithm problem for that particular prime.

I've also read that SRP is very similar to Diffie-Hellman, and has common groups (see RFC 5054). Can the attack be mounted on SRP as well?

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In the introduction of the Logjam paper, it is stated that

After a week-long precomputation for a specified 512-bit group, we can compute arbitrary discrete logs in that group in about a minute.

So it seems that what it actually does is attack the discrete logarithm problem, so any discrete-logarithm-based system which uses a common prime should be equally vulnerable. In particular, in Section 3.2 of RFC 5054, it is stated that

An attacker who could calculate discrete logarithms % N could compromise user passwords, and could also compromise the confidentiality and integrity of TLS sessions. Clients MUST ensure that the received parameter N is large enough to make calculating discrete logarithms computationally infeasible.

Since RFC5054 specifies only one prime for each bit-length, it will indeed be vulnerable. Even though it does not specify a 512-bit prime, it does specify a 1024-bit one, which is considered vulnerable by the Logjam authors:

We estimate that even in the 1024-bit case, the computations are plausible given nation-state resources

Using a prime of at least 1536 bits should be sufficient to thwart such attacks, since there is no known "downgrade" attack on SRP.

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