I would recommend using the IActiveView.Extent property. You can cast an IMap to an IActiveView and vice versa. http://resources.arcgis.com/en/help/arcobjects-net/componenthelp/index.html#//001200000199000000
The Extent property will return an IEnvelope that has an X and Y Max and Min that you can use to calculate the center if you would like.
To get the IActiveView instance you can use the IApplication.Document property and cast the Document to an IMxDocument. You can then get an IActiveView using IMxDocument.ActiveView property. http://help.arcgis.com/en/sdk/10.0/arcobjects_net/componenthelp/index.html#//004900000054000000
Dim mxDocument As ESRI.ArcGIS.ArcMapUI.IMxDocument = CType(application.Document, ESRI.ArcGIS.ArcMapUI.IMxDocument) ' Explicit Cast
Dim activeView As ESRI.ArcGIS.Carto.IActiveView = mxDocument.ActiveView
Dim currentExtent As IEnvelope = activeView.Extent
//TODO:: Get Max/Min X and Y from currentExtent and calculate the center point.
Here is a gis.stackexchange post that describes taking a map x,y and converting it to a lat, long.
Extracting lat/long from screen when using British National Grid projection in ArcGIS Desktop?
In particular, follow these steps-
I'm not that familiar the the British National Grid, but it appears
that a datum transformation is required when projecting to WGS84.
So, use ISpatialReferenceFactory.CreateGeoTransformation to create the
appropriate datum transformation, and
ISpatialReferenceFactory.CreateGeographicCoordinateSystem to create
the target (WGS84) spatial reference.
Finally, cast the point into IGeometry5 and call ProjectEx, passing
the target spatial reference and the geotransformation. IPoint.X and
IPoint.Y should then represent longitude and latitude of the point
where the user clicked.