I am going to guess Cricky81 that the "$30" interface you refer to is the Behringer UCA202/222?
If so then yes, it will work really quite well, had several such converters in the past. There are caveats...
The devices are 16bit only and so the noise floor is at best some -80dB fs. This is however well below the likely noise in a microphone recording since even with the best mic pres in the world, 'room noise' is not likely below -70dB fs.
The other problem with the UCAs is low headroom. One volt rms is all they can take and you would be well advised to stay a good 6dB below that. Even though the mixers outs are RCA I suspect they are still at the full fat +4dBu OP level and so you will need to keep the main fader well down.
Far better is a couple of pots* on the RCA outs to set an appropriate level into the 202 whilst having a good level from the mixer.
Pity you are in $ land as I would happily post you all this kit. I have no need anymore (and better)
*You could use a stereo potentiometer 10,000 Ohms 'log' taper but individual 10ks give you channel balance setting. You don't even need to put the pots in a tin as a rule. Set then gaffer tape to stop accidental buzzes.
Yup, just pulled the manual for the Berry 1204 Eu mixer and the RCA (CD) out run at +22dBu max. Bit silly really for outputs that are intended to connect to 'domestic' kit such as hi fi amps and cassette machines that run at neg ten.
Dave.