Ways to Fix Antenna Violations

Add antenna diodes near the gate.
Switch to a higher metal layer near the gate.
Insert a buffer near the input gate if that path is not timing-critical.
Connect the antenna-violating net to a buffer's input pin while leaving the output pin floating or on a dummy load.

KEY Fix antenna violations with diodes, higher-layer jumps, buffer insertion or connecting the net to a buffer input.

Fixing DRCs Near Tape-Out

When cell density is high in that area:

Identify the non-critical nets there with comfortable positive slack (more than about 150ps) and ECO-route only those incrementally with timing-driven and SI-driven options turned off (delete the nets and their global routes, then reroute with route_zrt_eco) so the tool routes them away from the area.
Find buffers and inverters on non-critical paths in that area and downsize them to free up routing tracks and space.
Convert the multi-cut vias in that area to single-cut vias.
Perform area-based DRC cleaning.
Take the nets on the critical paths and incrementally reroute them on metal layers above the highest layer used in the block.
Cell padding or module padding can be applied, but doing it indiscriminately may hurt timing because it disturbs critical cells too.
As a last resort, trim PG straps by removing some vias without breaching the foundry IR-drop limit - but this is less preferable when density is very high.
Add guide buffers on non-critical nets crossing the DRC area and place those buffers away from it.

When cell density is low (assume the DRCs come from feedthroughs or nets crossing top-to-bottom):

Most of the above techniques do not apply because there are very few cells to work with.