This has been gone through before. The club can appeal any IC decision (on liability or sanction). It can do so for any reason under Rule W but it is inevitable that the IC's factual findings will be hard to unwind - if they find Mr X an unreliable witness, that is it. The Appeal is not going to recall Mr X for re-cross examination - City will be stuck with that finding. Where the IC has got the law wrong or added extra sanctions even though, say, the PL never invited a particular finding on that point (effectively what happened in Everton 1), that can be challenged. Likewise, a wholly disproportionate sanction could also be challenged. In that case, like Everton, you could expect City to throw every argument about sanction you can think of at the Appeal Board but probably still have to accept the IC's factual findings.
The Rule W appeal decision is final. Save for Rule X.
Rule X.37 says: Subject to the provisions of sections 67 to 71 of the Act, the award shall be final and binding on the parties and there shall be no right of appeal. There shall be no right of appeal on a point of law under section 69 of the Act. In the event that a party to arbitration under this Section X challenges the award, whether in the English High Court or any other forum, it shall ensure that the League is provided with a copy of any written pleadings filed and/or evidence adduced as soon as reasonably practicable after their/its filing.
In simple terms this is a very limited final appeal (CAS not possible) - Sections 67, 68, 70, and 71 of the Arbitration Act 1996 allow challenges to an arbitration award based on lack of jurisdiction (s.67), serious procedural irregularity causing injustice (s.68), procedural rules and time limits for bringing such challenges (s.70), and jurisdictional objections by non-parties affected by the award (s.71). It is really unlikely any of those could be said to apply to City's case.
Note that even if the Rule W got the law completely wrong, the parties are stuck with it