Are you sure you meant to say the existing drive was 8Gb??
I will work OK with Win98.
However, it is possible your BIOS may not recognise 80Gb. If you try it, and it won't, ask again because it can almost always be corrected.
A second drive will not slow the system down at all - in fact with a 7200 spin speed, your system will run faster.
But why did you say a second 80Gb drive when you said the first was 8Gb??
I can't tell if your system has a second IDE cable - you will have to look to find out. If you need a second cable, be sure to get an 80wire cable, and use it as the Primary cable - unless you alraedy have one, in which case get a 40 wire cable and use it as the Secondary cable. Note the connectors are the same, it's only the number of wires that is different - count them if you aren't sure.
Normally you will install the new drive as the first drive (Master) on the Primary IDE cable, and the old drive as the Slave on the same cable. Your optical drive(s) should go on the Secondary IDE channel.
Is it a long process?
Physically it should take you about 30 mins maximum. Transferring the drive contents should take another 30mins or so, with the right software - ask about this if you are not sure.