java: How can I do dynamic casting of a variable from one type to another?

You'll need to write sort of ObjectConverter for this. This is doable if you have both the object which you want to convert and you know the target class to which you'd like to convert to. In this particular case you can get the target class by Field#getDeclaringClass().

You can find here an example of such an ObjectConverter. It should give you the base idea. If you want more conversion possibilities, just add more methods to it with the desired argument and return type.

Yes it is possible using Reflection

Object something = "something";
String theType = "java.lang.String";
Class<?> theClass = Class.forName(theType);
Object obj = theClass.cast(something);

but that doesn't make much sense since the resulting object must be saved in a variable of Object type. If you need the variable be of a given class, you can just cast to that class.

If you want to obtain a given class, Number for example:

Object something = new Integer(123);
String theType = "java.lang.Number";
Class<? extends Number> theClass = Class.forName(theType).asSubclass(Number.class);
Number obj = theClass.cast(something);

but there is still no point doing it so, you could just cast to Number.

Casting of an object does NOT change anything; it is just the way the compiler treats it.
The only reason to do something like that is to check if the object is an instance of the given class or of any subclass of it, but that would be better done using instanceof or Class.isInstance().

Update

according your last update the real problem is that you have an Integer in your HashMap that should be assigned to a Double. What you can do in this case, is check the type of the field and use the xxxValue() methods of Number

...
Field f =  this.getClass().getField(entry.getKey());
Object value = entry.getValue();
if (Integer.class.isAssignableFrom(f.getType())) {
    value = Integer.valueOf(((Number) entry.getValue()).intValue());
} else if (Double.class.isAssignableFrom(f.getType())) {
    value = Double.valueOf(((Number) entry.getValue()).doubleValue());
} // other cases as needed (Long, Float, ...)
f.set(this, value);
...

(not sure if I like the idea of having the wrong type in the Map)