A relative client ID (i.e. not starting with :) is relative to the current parent NamingContainer component. The <ui:repeat> is also a NamingContainer. So the render="panel1" is looking for the component within the context of <ui:repeat>. This isn't going to work. An absolute client ID (i.e. starting with :) is looking for the component within the context of view root. But you've it inside a <h:form> -which is in turn another NamingContainer component-, so the render=":panel" is also not going to work.
The following should work, with prependId="false" removed so that you can refer it:
<h:form id="form1">
  <h:panelGroup id="panel1">
    <h:dataTable/>
    <ui:repeat value="#{bean.page}" var="page">
      <h:commandLink value="#{page}">
        <f:ajax execute="@form" render=":form1:panel1" listener="#{bean.page}" />
      </h:commandLink>
    </ui:repeat>
  </h:panelGroup> 
</h:form>
By the way, if you actually want to render only the table, then you should be doing this:
<h:form id="form1">
  <h:panelGroup>
    <h:dataTable id="table1" />
    <ui:repeat value="#{bean.page}" var="page">
      <h:commandLink value="#{page}">
        <f:ajax execute="@form" render=":form1:table1" listener="#{bean.page}" />
      </h:commandLink>
    </ui:repeat>
  </h:panelGroup> 
</h:form>
Update: as per the comments, it turns out that you have changed the JSF default NamingContainer separator character from : to _ by configuration. In that case, you need to use _ instead of : in the client ID selector.
<f:ajax execute="@form" render="_form1_panel1" listener="#{bean.page}" />