You have taken over as PM on a project that is behind schedule and over budget. What will you do to recover the situation?

It is a definite tough situation. There could be problems with resource optimization, or even improper resource selection, along with poor control measures over the project. Gain confidence of the team and management. Design a best execution plan along with team for preventing further damages. Analyse what went wrong previously, and try not to repeat the same.

Work on the Longest tasks on critical path and squeeze and on the remaining tasks consuming the majority of the cost and find alternative solutions

It clearly states that the PM was laid back and did not allocate nor utilize the resources properly. 

The only way out is, look where you went wrong in the process or which part of the process is not looked into properly which is creating the delay. Work on those processes and parts of the project, putting more time into getting it done effectively and efficiently by overlooking the work and finishing the project within the scheduled time and budget.

If the PM and team fails to do so, all are responsible for the delay and not meeting the requirements of the project.

To prevent such instances in the future, a complete RCA of the matter should be done and a solution to avoid these scenarios.

This depend of the phase if it is the engineering phase or procurement phase , or construction phase , to recover the schedule for the engineering and construction this could be done by using more manpower ,this depend on the budget , and the priority of the activities in delay  but for the procurement , a risk analysis shall be done regarding the delay if it possible to accept other resource or to take the risk for the delay , regarding the budget ,   you have to lookout for the necessity and the priority of the activities as well , if it possible to cancel or not , because most of the projects comes over budget because of change orders for which you have no thing to do 

Depending on the time constraint you may be limited to what you can do the best would be to salvage what you can without causing any more  delay or obstruction. Prioritise the tasks in order of urgency and importance to identify what needs rectifying first. Focus on cutting down costs in unnecessary departments, reduce the expenditures, find alternatives to specific procedures. Consult and inform senior management of what outcome may result of this situation so that they are aware.