I can get
My thought process is
I can take 5 out of every 11 consecutive cards. If I take cards (1, 3, 4, 6, 9), and then (12, 14, 15, 17, 20), etc. I will have taken 915 of the first 183*11 = 2013 cards. I can then take (2014, 2016, 2017, 2019, 2022) to bring the total to 920.
I believe this is the maximum because
choosing a card eliminates four other cards from being chosen (ignoring the edge cases). An optimal strategy would seem to be picking subsequent cards which "duplicate" as many eliminated cards as possible. So for example, if we pick the card numbered 10, we've eliminated cards 3, 6, 14 and 17. For the next choice, we'd like to pick something that also eliminates as many of (3, 6, 14, 17) as possible, to minimize the number of new cards we're eliminating. It's easy to see that we can choose a number that eliminates two of the same cards as our previous choice by picking a number 3 greater, e.g. picking card #13 in this case, since it also eliminates cards 6 and 17.
It's also easy to see that we can repeat patterns of 11 cards. So by picking every third card, we end up with four cards in a group of 11, plus a gap where we can choose a fifth card.
I've made two pictures to help illustrate what's happening.
This shows the first step of choosing every third card (blue spaces are cards we've chosen, pink spaces are cards we cannot choose by virtue of the cards we've already chosen).
This shows the completed pattern once we've chosen a fifth card.
And since 2022 is 9 mod 11, we want to align the pattern of choices so that all 5 cards are in the first 9, so that we can get 5 of the last 9 - i.e. the pattern pictured in my links above might have us choose cards 1, 2, 4, 7, 10, ... but then we would only get 4 of the last 9, leaving us with a total of 919 instead of the optimal 920.