FastCGI is (I believe; slightly rusty but going from memory) a C library implementing its API that lets a web server (e.g. mod_fcgi in apache) start up a persistent process which it will send requests to for handling. Theoretically one could write your own implementation of the API in perl (there's Net::FastCGI on CPAN, but I have no experience with it and since it looks that module hasn't been updated in 7 years . . .), but if your perl install is so stripped down (one might even say crippled) that it doesn't have strict you're probably not going to have a good time of it.
What I'd try would be seeing if you can instal a more complete perl (a quick googling shows OpenWRT splits perl into an extremely strangely large number of subpackages so keep adding those) and the C FastCGI library and try installing the normal FCGI module from CPAN then.
The cake is a lie.
The cake is a lie.
The cake is a lie.