It can be solved so long as your date is simple. That is, if you have three fields, one for month, one for day, and one for year, it should not be difficult. If you have a date field, though, it gets tougher.

The basic field is an IF field. You have to identify your month field in some way. I would suggest using either Mapped Content Controls or legacy form fields for your drop-down. Content Controls and legacy fields should not be used in the same document, you should choose one or the other.

The advantage of the legacy controls is that you can check the "calculate on exit" box in properties and any IF fields based on that control will update automatically when the user tabs out of the control. A second advantage is that it is easy to give a legacy control a bookmark name. A disadvantage is that the document must be protected for "filling in forms" which disables ability to use many other Word features. Another disadvantage is that these are "legacy" controls and at some point Word may stop using them.

The advantage of content controls is that it can be "mapped" so that you do not need bookmarks to reference it. Another advantage is that such controls are likely to be supported in Word longer than the legacy controls. The document does not have to be severely restricted as far as editing goes.