Working with Google Maps/ArcMap lat/long

Due to various vagaries in the ArcGIS software, I'm going to tell you to use different coordinate reference systems. That will let you access the predefined transformations and hopefully make your life easier.

Use WGS_1984_Web_Mercator_Auxiliary_Sphere instead. This one uses WGS 1984 for the geographic CRS (aka datum). The one you're using, WGS_1984_Web_Mercator, uses a sphere (GCS_WGS_1984_Major_Auxiliary_Sphere) and there are no predefined transformations for it.

Next, use Monte_Mario_Italy_1 which is the same, just a different name. Pick Monte_Mario_To_WGS_1984_4 for the transformation.

Or if you want to use Roma_1940_Gauss_Boaga_Ovest, you'll have to define a custom transformation that's a copy of Monte_Mario_To_WGS_1984_4.

FROM roma 1940 TO wgs84
Method: Position Vector
X axis translation: -104.1
Y axis translation: -49.1
Z axis translation: -9.9
X axis rotation: 0.971
Y axis rotation: -2.917
Z axis rotation: 0.714
scale difference: -11.68

If you define a custom transformation in ArcMap, you may have to define it FROM wgs84 TO roma 1940. If so, change the signs on all parameters.

Datum transformation are never fully accurate, which could explain a small shift but not a difference of 30 to 40 m.

A difference of 30 to 40 m is more like an error due to the absence of datum transformation, or it could be an error in the dataset that you use for the relative evaluation of your position (Google registration can be wrong in some places of the world).

In order to solve this issue, you should try to contact the person who provided the lat/long coordinates in order to know in which CRS he was working. Then you should check your transformation used in ArcGIS to make sure hat it is corectly used. An error of transformation would result in a systematic error, but of course this is only visible if you have only one point. Another way to check for this is using an independent projection tool (e.g. http://twcc.free.fr/ ) and compare your coordinates with ArcGIS

EDIT :

By looking at your image, I see that your both coordinate are dd°mm'ss.0'' It therefore seems (99% chance) that you don't have the decimal values. Hence your precision is thus about 30 m in each direction. If the coordinates have been truncated, the error is up to 30*sqrt(2)= 42m. If it has ben rounded, this maximum error can be divided by 2.