I understand "apostolicity" to be the characteristic of the church in which
(1) it is faithful to what the apostles taught
(2) it proclaims the gospel the apostles proclaimed
(3) it continues the ministry of the apostles, that is the ministries of teaching and preaching the truth of Jesus Christ and founding churches based on that truth.
Another way of putting this is that the church is apostolic when it continues what the apostles did and said in order to enlarge (through church planting) and maintain the church of Jesus Christ.
The question of "apostolicity" generally arises because when we think "church" we are invited to think what "church" means. Can it mean whatever we make of it? Is it bound to mean what it has always meant? (If so, why?) Is some kind of continuity (apostolicity), comprehensiveness (catholicity), constraint (holiness) and community (unity) critical to church being authentically church?
The question particularly arises when as church we consider making decisions which push against, if not break through the "unity, apostolicity, catholicity, holiness" of the church, or, if you prefer, break through the continuity, comprehensiveness, constraint and community of the church.
Alternatively the question of apostolicity might arise when a teaching pushes against the same boundaries of what it means to be the church. In the letter to the Ephesians in Revelation 2:1-7, we read:
"I know you that you cannot tolerate evildoers; you have tested those who claim to be apostles but are not, and have found them to be false. ... But ..." (2bc, 4a)
Even at the end of the first century, the question of a true or a false apostleship was encountered by the church. (I included the "but" at the beginning of verse 4 as a reminder that when a church has rounded up and shipped out false apostles it is not immune to the Lord's judgment as to it's true spiritual health!).
Now apostolicity has some variations in emphases as different Christians in different traditions have reflected on what apostolicity means. Some place emphasis on content of the faith and its continuity: we are apostolic when and only when we continue to the content of the faith of the apostles. Others emphasise the leadership aspect of apostolicity: we are apostolic when we continue the office of the apostles (i.e. via bishops). On this emphasis, the content of the faith also matters: bishops are chosen who will maintain the teaching the apostles and lead church's the proclamation of the gospel.
Naturally, Anglicans have the best of all apostolic worlds [ :) ].
Revved up by the Reformation we value the (re-formed) content of the faith once delivered and we refused to rid ourselves of bishops. But that claim (made slightly in jest, slightly in seriousness) is not likely to cut much ice with three major alternative views on apostolicity.
The Orthodox propose that they have the purest adherence to the apostles: bishops, strict adherence to (genuinely universal, undivided church) ecumenical councils' interpretation of the apostles' teaching, and continued use of the first Scripture of the apostles (the Old Testament in Greek).
Roman Catholics propose that they have the safest adherence to the apostles: bishops who teach what the apostles teach and a bishop of the bishops (pope) to secure that teaching by both preventing heresy gaining ground (e.g. ability to sack any bishop around the globe who steps out of theological line) and by only promulgating new teaching after a long, careful, well-tested process of review (possibly taking centuries), including an assurance that the whole (Roman) church is united behind the new teaching.
Reformed churches (in the sense of those churches formed during the Reformation and subsequently (e.g. Methodists) but who eschew bishops) propose that bishops are as often the problem as they are the solution to strengthening apostolicity. They would point out to Anglican and Lutheran churches that individual "rogue" bishops are not easily silenced; and ask of the Roman church, what if the Pope goes rogue?. It is happened. Some say it is happening. I digress!
Although I am not personally aware of what a "Reformed" critique of Orthodoxy might look like, I can imagine it might point to the divisions within Orthodoxy, including the division between Eastern Orthodox and Oriental Orthodox as signs that a claim to purity of adherence to the apostles is no guarantee of unity of adherence: patriarchs and bishops can get in the way. Instead, the Reformed approach to apostolicity, as I understand it, emphasises the content of apostolic teaching, and permits all and sundry to advance it, defend it and debate it in order that individual heretics are efficiently condemned and/or sidelined (with no need to pay respect to their episcopal office let alone to their professorial office) while other champions of apostolicity are widely praised and their writings propagated.