Get member to which attribute was applied from inside attribute constructor?

Attributes don't work that way, I'm afraid. They are merely "markers", attached to objects, but unable to interact with them.

Attributes themselves should usually be devoid of behaviour, simply containing meta-data for the type they are attached to. Any behaviour associated with an attribute should be provided by another class which looks for the presence of the attribute and performs a task.

If you are interested in the type the attribute is applied to, that information will be available at the same time you are reflecting to obtain the attribute.

It's possible from .NET 4.5 using CallerMemberName:

[SomethingCustom]
public string MyProperty { get; set; }

Then your attribute:

[AttributeUsage(AttributeTargets.Property)]
public class SomethingCustomAttribute : Attribute
{
    public StartupArgumentAttribute([CallerMemberName] string propName = null)
    {
        // propName == "MyProperty"
    }
}