"COLD START as deputy subproject manager in an already running project"

Some general guidance -- Assess the situation:
How is progress according to the schedule (what's due, what's late)?
Where are the bottlenecks?
Is there an issue list? Is it prioritized? Who is working it?
Who are the key team members you can rely on to keep things moving while you get up to speed?

Taking the time to understand and apply trusted methods of project management has the power to create more efficiency, organization, and output across your organization. Theories and processes for effective project management abound, but all are based on a few basic tenets. Wherever you’re working on, you need to know:

Where are we going?
How are we going to get there?
Do we have everything we need?
What impediments will we encounter, and how will we remove them?
How are we going to measure our progress?
If every project plan takes into consideration these five questions, success will be much easier to define, measure, and achieve.

Ulrike, I would suggest to start with DMAIC approach.

Firstly - Congratulations. The fact that your superiors dump an important project into your lap says that they have confidence in your abilities. My favorite tools would be:

Checklist - use it to determine the overall health of the project. What milestones has been completed, are coming up, were missed, deliverable done/to be done etc.This can all be extracted from a good schedule, if this is not the case the checklist will give you the opportunity to fix it.

Mind maps - use it to get a visual overview of the scope. I like using colors and icons on mine to indicate priority, state etc.

Interviews - talking to team members and stakeholders to get a feel of the general mood and where you are at. It also give you the opportunity to communicate your involvement and approach.

Complexity matrix - for each project I would define a list of parameters that I can use to gauge the complexity nature of the project i.e. skills requirements, technology, # of stakeholders etc. This gives you an indication of where to focus your attention and it is also a living matrix that should change (for the good) though the project life cycle. If it does not then it means we are not learning and adapting.