What you call "a small Orange tree" is known as a Dwarf Citrus tree, so an Orange Dwarf tree in your case.
As for when to prune, well you can prune a citrus tree throughout the year.
Citrus trees respond very well to pruning, it will encourage new growth. From what I see (I didn't get the whole picture) you only need to prune in the places where you see the white lines, more or less.
Once pruned you will:
1. Prevent branches from growing too long (those are the branches that later give the tree a round bushy look)
2. See new growth of young strong leaves under the pruning point.
One of the basic rules of pruning is that it encourages new growth.
As I mentioned earlier, i didn't receive the entire picture, but I would recommend the following:
Once pruned, place the plant in a sunny spot where it can get maximum sunlight. If that is already the case – great . ( and ignore this last comment)
I see in the picture that the tree is planted in a pot, with some kind of nylon bag inside it. I don't know if water comes out the bottom when you water the citrus, but please make sure that there is a drainage hole at the bottom.
For the record, a drainage hole is a must for any plant. water should always come out pf the bottom as you water your plants. Any plant. Any pot.
Regarding the pot, although it is a dwarf tree, you should consider replacing the pot with a larger one, it will allow more room for the roots to develop. If you do decide to replace the pot, pleasewait until spring, it will make it easier for the plant to adjust to its new home :)
As for the water regime, if you have an automatic drip irrigation system, I suggest you close it until mid- spring. Check the soil by touching it with your finger once a week and only add water when it feels dry. How much water? just feel it! My rule of thumb is to add water until I see it drip out the bottom.
One last tip: It is a common mistake to give water every time we think something is wrong with our plants, but in most cases, we're doing them more harm than good. Over watering only makes it harder on them.
For more information on watering, take a look here: http://www.handpruning.com/how-to-water.html
I hope I covered everything you need, but if there is anything else I can help with, please send me another post. And I'd love for you to keep me updated on your citrus dwarf as you start the results of your pruning over the next few weeks.
shrub removal - Of course having a gardener do it is easier, but getting down and dirty is more fun and will give you loads of satisfaction. You can DIY the removal of old shrubs!