TL;DR: there is no distinction between arrival area and departure area, so it might be possible. However, there will be logistical and possibly legal problems. I don't think it will be worth the risk.
As of the logistic part of the question, this is absolutely no problem. At Changi Airport, the arrival and the departure zones are linked together and you won't have any problems to access the gate once you are through immigration. You won't even need to go through security check as those only take place at the gate. For more details of how to get from the arrival customs to the gates, you can see their online map of T3. As @uncovery noted in the comment, it might be quite hard to find your wife, as arrival gates are usually not announced. She will have to pass through one of the two escalators located at Transfer A and Transfer B on above map to get to immigration, but if you don't know which gate she is arriving at, you won't be able to know at which side to wait for her.
I would urge you however to consider that this is going to cause some problems to the airline, as they will have to call for you, since you will do the check-in and then not board the plane, this will make for instance a potential early departure of the plane impossible. If you do so, I would suggest that you book a flight departing well after your wife arrives and then go back to the check-in counter to tell them that you are not actually flying.
I can't help you with the question if this is legal. You should see jpatokal's answer for some examples of how this might (probably will) go wrong.
Note that the immigration process at Changi airport is among the most straight forward ones I have seen. If you want to help your wife, help her to prepare all the details she has to fill in for the D/E form and make sure she has all the documents for her visa if she needs one and she should be fine. I have passed through the Singapore customs many times and rarely ever had to talk to them.
Also note that there are no budget airlines departing from T3, so in order to get a cheap one-way you would have to check-in from one of the other terminals and then transfer, which is possible, but will only make it more annoying. If you are really scared, that your wife will not make it through there alone, I suggest you do consider the Passenger Meeting Service which will save you all this potential trouble and seems to be cheaper than buying a flight.
Update: You might be interested in this article, quoting the police on how it is illegal to enter the transit area without wanting to travel.