Welcome to old N64s. i strongly recommend not blowing into the cartridges or the cartridge slot as we used to do as kids - as it turns out, all it does is just leave tiny bits of spit on the connectors and ends up rusting them
A trick I found was just to plug and unplug the cartridge several times before switching the unit on. By doing this, you're basically scraping off the oxidized layer from the cartridge's PCB tracks, and perhaps even the connector inside the 64 - shiny surfaces will usually always become oxidized (except for gold I think). I sometimes had to do it 20-30 times (while the unit was off of course) - but it always seemed to work.
Forget blowing - it won't help a cartridge that hasn't been used for 10 years. You're probably going to make the oxidization worse.