Slot 1 CPU's do demand a heatsink AND FAN. In most cases, companies like Dell and Compaq used their own custom cooling solutions consisting of one big heatsink (larger than the slot chip itself) so that one part with machined holes for air stuck out over the slot cartridge and was cooled by the airflow from a bigger passive fan nearby, OR the companies used just a big heatsink maybe 2x the size of the standard Intel unit and also had a passive fan near the chip cooling it.
It depends on the P3 also as to if you can get away with running it without a fan. There are 2 types of P3-600, the coppermine version akin to the socket 370 type, or the older Katmai version which still has external cache chips and is made using a larger die process and hence runs hotter.
The Katmai will not run without a fan at least passively blowing onto the cooler if it has a very larger heatsink. I know this as I had a P3-600 Katmai and it ran hot (although still not a match for todays CPU's) it just generates too much heat.
The coppermine version will run cooler but I can't honestly say how much cooler. I have a P3-800 slot 1 coppermine chip and even when running prime95 it gets lukewarm and thats with the standard intel heatsink and fan unit.
How to tell if its a Coppermine or Katmai chip? Apart from looking up the S-spec on Intel's website, the chip has an identification label on the top of the slot cartridge (opposite the connector side). Its broken down into the following format:
600 / XXX / YYY / Z.ZZ V(volts) S1
XXX will be either 512 or 256. There shouldn't be any other possibility. If its 512, its a Katmai. If its 256, its coppermine.
YYY is bus speed in MHz, either 100 or 133 will be there, but I'm assuming it will be 100.
Z represents the chip voltage. Katmai's use 2 or 2.05 volts, coppermines run at 1.6->1.7volts.
S1 denotes slot 1.
The only other issues you would face is if the motherboard supports the correct voltages for the coppermine chips or the 133MHz bus speed if its a 133MHz chip. Half the 440BX chipset motherboards don't support the coppermines at all well so check that out before you blow up the chip. The problem with the bus speed depends on the chipset also - the 440BX doesn't do 133MHz too well on some motherboards as the chipset doesn't support the correct AGP speed at that frequency so you would be forced to use a PCI graphics card. Other chipsets such as the VIA 694X, or the i815 handled these problems because they are a lot newer.