As in all of Micronesia, dogs can present a problem and though I didn't experience this personally on the island I was warned about it. When out walking, esp. alone, it's not a bad idea to have a stick with you to protect yourself. Also I was told (again this works in the rest of Micronesia as well) just the action of looking like you are picking up a rock to throw will back them off. The good news is there does not have to actually be a rock handy to pick up.
Water is extremely precious in Nauru, so be very conservative with your use of it. The only fresh water is in the stagnant Buada Lagoon, and there is no facility to transport this. The desalination plant, which used to remove the salt from sea water, has broken down, so now residents and the hotels are left with rainwater as their only source. This is stored in tanks but it can sometimes be a long time between rainfalls, resulting in dangerously dry periods.
There is plenty of bottled water to drink, but please, be very tardy with your water usage for cleaning purposes.
Nauru is a pretty safe country over all, with no weather hazards and not much crime, but here are a few things to bear in mind:
Stonefish - take care walking in the shallows or on the reef because these nasty little things sit on the ocean floor looking exactly like the ocean floor, until you step on them, when they inject you with potentialy lethal venom! Occurrences are rare, but be on the safe side and wear soles if you stray off the sand.
Other marine nasties - sharks and jellyfish hang out here but are not usually a problem.
Be careful of dangerous currents if swimming.
Fresh water can sometimes be in very short supply in Nauru so be considerate in your usage of it. If it runs out, you have to wait until next rainfall - and that can sometimes be a very long time coming.
Walking/bike riding at night - there is no streetlighting so either make yourself seen with a torch or be very vigilant. Drink-driving can sometimes be a problem.