Can you tell me which of the two readings come first in the printed Lectionary?
“First” doesn’t make sense here, because the proper of seasons (readings for the liturgical seasons of the year, the “seasonal” cycle) is distinct from the proper of saints (readings for saints’ feast days, the “sanctoral” cycle) and the two are in separate places in the Lectionaries.
The USCCB web site has an
online day-by-day Lectionary.
if the memorial for the Saint has a “proper reading”, then it “must” be used (per the
USCCB). Is that the case here?
I’m not entirely sure. I would
expect a “proper” reading for a saint (that is, a reading chosen for THAT particular saint) would be required, but a “common” reading (that is, a reading for a class of saints – martyrs, for example) would not be required and the priest could opt to use the “proper” of seasons instead.
Judging from the USCCB readings for June 10-12, the
first reading on June 11th (Acts 11) is a proper reading for St. Barnabas (since the first reading on the 10th and 12th is continuous from 2 Cor), but the Gospel reading is continuous with the day before and the day after, so the Gospel is probably the seasonal proper (rather than a sanctoral proper or common).
This is a rather particular situation. June 11th is the Memorial of St. Barnabas, yes. But the Thursday after Holy Trinity Sunday is (supposed to be) the Solemnity of Corpus Christi. However, in the US, Corpus Christi is transferred to the following Sunday. So the USCCB online Lectionary reflects the First Reading for St. Barnabas but the Gospel reading for the Thursday in Ordinary Time. The CatholicCalendar web site you linked seems to be getting its Gospel distinctly from the sanctoral cycle, and I am quite sure the Gospel Reading (from Matt 10) is a “common” Gospel reading for saints like Barnabas who are close associates of the Apostles.