To access Office 365, there are some prerequisites that a client and the network the client is connected to must meet for Office 365 to work correctly.
TCP 443 – O365 Portal, Outlook, OWA, SharePoint
TCP 80/443 – Azure AD Sync Tool, Mail migration tools, Exchange.
TCP 25 – Mail Routing
TCP 587 – SMTP Relay
TCP 143/993 – IMAP Simple Migration Tool
TCP 995 – SPOP3
Further ports required by Lync
TCP 5223 – Lync Mobile clietn push notifications.
PSOM/TLS 443 – Lync Online outbound data sharing.
STUN/TCP 443 – Lync Online outbound audio, video, app sharing.
STUN/UDP 3478 – Lync Online outbound audio and video sessions.
UDP 20000-45000 – Lync to Phone outbound.
UDP 50000-59000 – Lync outbound audio and video sessions.
Office 365 Best Practices Analyzer – diagnose client connectivity. Requires Windows 7 SP1 (64Bit), IE9 onwards.
Access this from Tools menu in the O365 Admin center.
Office 365 OnRamp Tool – Check deployment readiness in on premises
Access this tool from https://configure.office.com