Use an acid based cooling system cleaner.Follow instructions but leave it in overnight.Most are based on phosporic acid and is a fairly mild mix and can be safely left in overnight and drained next day ( after another 15 minute run).Flush system well after draining.If still rusty/oily give it another dose and same procedure.
Or better still, find a mob that can flush it professionally using a wynns flushing machine or similar.
Once it has been flushed, use a rust inhibitor in the system.
Had previous owners used rust inhibitor in the system then you wouldn't be in the situation you are now.
I'd pay the money and get it flushed properly. A mix of chemicals and high pressure plus if the place is any good they'll check everything while they are at it.
Speed and Style........... One day I'll get the speed bit.