You might notice from the map above that the Oberoi do not show any road exit. They do not really seem keen for you to leave the hotel.
They make a great theatre of arrival - two cars at the airport, one for luggage one for you. These drop you at a jetty where you are taken to the hotel by boat. The bags go separately by car (the boat is plenty big enough for the bags, this is just how they do it).
On arrival at the jetty there is a golf cart for transfer to the front gate - good for the heat, not so great in the driving monsoon rains.
They told us there is 'nothing' near the hotel, this is in fact entirely untrue. If you leave the hotel you can walk straight ahead to the jetty or left will take you to the road entrance. The entrance is shared by the Trident (Oberoi) which in fact borders the Oberoi's grounds and has a staffed kids' club, we stumbled on it - they seemed happy for us to use it despite being at the Oberoi rather than the Trident, they also offered various activities including free camel ride or horse ride (elephant ride is 3000 rupees, which I would suggest is likely poor value cf. Dera Amer in Jaipur).
Immediately out of the main gate there are a number of souvenir shops and a few small hotels and restaurants. All down the road is built-up albeit that the (old) city proper is a couple of km away. After about 400 metres there is a roundabout. Turn left and there is a 'wine shop'. Beer here costs 50 rupees against 350 rupees (for a small bottle) in the hotel, you can buy various other liquor items here too. Just across the roundabout is an ATM.
If it is hot or you don't want to walk just use a rickshaw. To the city palace, about 15 minutes is 50 rupees. To the city proper, some 25 minutes or by rickshaw we paid 100 rupees.
Food in the hotel is good, but so is all the food we had outside. The prices - around 1250 rupees for a meat curry might seem high compared with outside but the value is not so bad as this is tax/service included and they also gave us two rice, daal, veg curry, a plate of assorted poppadoms, and Indian bread. Of course it is still much cheaper to eat out and somewhere like Ambrai has a million dollar lakeside view that Udaivilas cannot match (if much cheaper furnishings) there I think we paid around 250 rupees for a curry, 95 rupees for rice, around 35 for plain roti, plus tax on top. The daal at Udaivilas is outstanding but we had a better butter chicken in a random restaurant near the City Palace. Our saag paneer was otoh fantastic and much better than anything I've had in the UK, but then we subsquently had lamb saag in a couple of different restaurants outside and they were all very tasty too.
The food here is curry after all, it's not based on lightness of touch and fresh vegetables but spices and thick sauces. Of course they will sell you Western food - we tried gazpacho (a little sour, needed onion) and a brie & proscuitto sandwich (nice proscuitto, brie predictably unripe) - but it's in my opinion utterly pointless for me to fly from London, home of dozens of fine European restaurants with appropriate seasonal produce and native chefs to provincial India and hope to come close to what I can eat within a few minutes of home.
So yes, stick to curry.
Regarding the hotel food we saw the same chef at dinner and breakfast, I would accordingly recommend that the food in the Udaivilas is very nice but otoh so is all the other food I've eaten - your extra rupees buys you better service and crockery above all else.