Capturing the result(print statements) of SP_UPDATESTATS
I would use the free Ola Hallengren maintenance solution to update statistics.

See Example C from the above link, where you can update the statistics in all user databases via: -

C. Update statistics on all user databases

EXECUTE dbo.IndexOptimize
@Databases = 'USER_DATABASES',
@FragmentationLow = NULL,
@FragmentationMedium = NULL,
@FragmentationHigh = NULL,
@UpdateStatistics = 'ALL'


You can also edit this for your needs to only do an individual database.

The logging you require can be achieved by specifiying the LogToTable parameter.

As a final point. Remember you don't have to update statistics if you are rebuilding your indexes. Rebuiliding your indexes during your weekly maintenance window might be your best option. Again you can use the Ola Hallengren solution for this.