In my opinion, if you have a large database that you can't export to flat files then you're probably going to have trouble making the conversion. Dumping data and reloading from files is fast.
An alternative would be to write a Perl script that read from the Sybase database and inserted into the MySQL database a record at a time. For a "lowly machine" this could be a real performance problem with a large database.
Not knowing the specifics of your situation (DB size, disk space, etc), the only other solution that I can think of would be to transfer the database in chunks. Move part of a large table to a temp table using a "Select Into" statement. Download the temp table to a flat file and then upload into MySQL. Drop the temp table. Change the "Where Clause" of your query and rerun for the next subset of the table. It might be possible to automate this process using Perl as well but for a one-time conversion it might be faster just to do it manually. Again it depends on your situation.