The 24-year-old Englishman was a free agent after leaving Kilmarnock in January and has signed until the end of the season.
Russell discovered this week that he had suffered a fractured fibula and will be out for at least four weeks.
United manager Jackie McNamara said: "With Johnny Russell's injury, we are short on forwards and when I heard Rory was available I moved quickly to bring him in for a trial.
"He has shown me he can offer us something with his energetic and busy style so I am delighted to have him sign a contract.
"He is fit enough to be considered for action immediately and I think he is someone the United fans will quickly grow to enjoy.
"He offers us a different presence from what we have available currently and he is keen to show he deserves a longer deal so this deal makes sense for both parties."
Boulding only made one start and three substitute appearances for Killie after signing last summer from Livingston, where he scored 12 goals last season.
The former Mansfield, Bradford and Accrington player said: "After being offered the chance to come in and show my worth here, I am delighted to sign this contract which gives me a further opportunity to win a longer term deal at the club.
"The ability within this squad is excellent and I want to be part of that.
"The manager has a vision of how he wants to take this club forward and to be offered the chance to be involved is something I am looking forward to.
"I am sure the United fans will enjoy my style of football and I know that we have a great opportunity to have a magnificent end to the season and I will do my utmost to ensure that."