Not directly. Assembler code is the least portable (well, apart from keying in raw hex machine codes).
You can probably do it in C, if I get my Arduino to generate VGA video using C, you should be able to generate audio at a reasonable rate.
Judging by a quick glance at the assembler code he has the various waveforms (like sine waves) as table data (to save generating it at runtime).
I have to say though, that unless you have a lot of time (he said he spent months on it) it might be easier just to get his chip.
If you don't want to do that, you would need to be able to understand PIC assembler enough to make out what he is doing there, and convert to either Atmega328 assembler (no simple task) or rewrite in C.