There are a few ways:
1. Dupe (cmd-D or ctrl-D) the layer, leaving the dupe selected, then alt-drag (opt-drag) the new layer from the project window to the dupe layer. The dupe layer will be replaced, but its keyframes and effects will be left intact. This doesn’t work with text layers.
2. Copy the effects from the old layer’s effects palette, select the new layer and paste. Not all settings will survive the paste — so check.
3. Select the effect keyframes from the old layer, copy and paste to the new layer.
4. Select the old layer’s animated properties (select the property names in the timeline), then animation>save preset, name it and save it. Next select the new layer, and animation>apply recent preset — the one you just made.
That should get you going — option 1 might be the first to try.
Steve