Twig Template print field - if not node number

Yes. Sometimes themes will drop documentation when they override a template, but you can always check the original template file for the available variables and other info relevant for working with the template. If you check the node.html.twig template provided by the node module in core, you'll see the following documentation at the top of the file.

/**
 * @file
 * Default theme implementation to display a node.
 *
 * Available variables:
 * - node: The node entity with limited access to object properties and methods.
 *   Only method names starting with "get", "has", or "is" and a few common
 *   methods such as "id", "label", and "bundle" are available. For example:
 *   - node.getCreatedTime() will return the node creation timestamp.
 *   - node.hasField('field_example') returns TRUE if the node bundle includes
 *     field_example. (This does not indicate the presence of a value in this
 *     field.)
 *   - node.isPublished() will return whether the node is published or not.
 * 
 * ... continued ...
 *
 */

Note the relevant info here:

The node entity [is available] with limited access to object properties and methods. Only method names starting with "get", "has", or "is" and a few common methods such as "id", "label", and "bundle" are available.

This means you can access the node id with node.id.

So, you can print your field on nodes other than your specific node by wrapping your field in a conditional like:

{% if node.id != 45 %}
  {{ content.field_big_image }}
{% endif %}

or

{% if not node.id == 45 %}
  {{ content.field_big_image }}
{% endif %}

Either choice should produce the same result.