Internet
Internet in Nice is usually high speed fibre, and if your building does not yet have it then it will soon. High speed Internet usually costs around €30/month, and if you want TV with hundreds of free channels and telephone with free calls to land lines all over Europe and the USA then this tends to be €40/month. The main providers to look at are:
Electricity
For a one bedroom apartment used as a holiday rental, a rule of thumb of €40/month is reasonable. Though the government has capped energy prices at 4% increase per year, this could have changed by the time of reading as could the rate of inflation. Other things that can affect your electricity bill are the efficiency of your hot water tank, air-conditioning, whether you offer kitchen appliances such as oven, dishwasher, etc.
Water
The city of Nice is built on a river and has its own water treatment plant, and water prices are very low here. Bank on €20/month. Most buildings here meter the residential consumption and then the Syndic read the internal meter outside each apartment and add it onto your monthly Syndic bill.
Syndic
This covers your buildings insurance, cleaning of the common areas, minor repairs and maintenance, keeping the lift in good working order, amongst other things related to the ongoing running of the building. The amount is fixed each year by the annual residents meeting, and every years you are provided with a breakdown of where all those costs go.
In the Old Town, the monthly fees tend to be around €80 but can be as low as €20 or as high as €120 depending on how many apartments in the building and the unlikely event you manage to find a lift. In the city centre it tends do be around €150, dropping to €80 if there is no lift and you are in a less desirable area up to €300 if you are in a historic building with multiple lifts and with a concierge.
Insurance
Typical insurance that has full coverage of everything you need, including contents insurance, are around €300/year for an average one bedroom apartment in a good area. The insurance companies also offer a "legal assistance" package for usually €5/month but generally this is not needed.
Taxe Fonciere
Generally fixed by area then by the size of your apartment. For an average one bedroom in the Old Town it will be around €600 per year, and for a nice two bedroom in the city centre it will go up to €1200 per year. If you are not sure yet what you are going to buy then €900 is a pretty good average price.
Not included here
There are two main other costs that you should put money aside for. The first is wear and tear. If you rent via a responsible management company which filters guests, then this can be surprisingly low. However if you want to drop prices low to fill the apartment as much as possible, then the lower paying customers that often enjoy the very wide variety of fantastic bars Nice has to offer, then some of those savings may go into early repainting of walls and repairing bathroom fixtures.
The second is that every 30 years or so the residents will vote to repaint the front of the building, or the stairwell which tends to not cost very much. An average I have seen is around €8,000 for repainting the exterior, though I have seen a range from €4,000 to €16,000 depending on the size of the building, if it is on a corner so has an extra facade, how many residents in the building are sharing the cost, and if it has a special listed status. In my personal opinion this is not an extra cost and should not be regarded as one. You bought the property more cheaply because of the state of the building, and it will be worth more after the work is done. It should be regarded as a capital reallocation into one of your most valuable assets. It makes it a better rental as well as increases the value of your asset so money well invested.