Short answer. Yes, a partial judgment can be executed before the rest of the case is resolved, but only through discretionary execution. The court must find good reasons, stated in a special order after a hearing, before allowing execution ahead of the final outcome of the whole case.
What the law says
A several, separate or partial judgment may be executed under the same terms and conditions as execution of a judgment or final order pending appeal.
Rule 39, Section 2 — Discretionary execution. Read the full provision →
Partial judgments follow the pending-appeal rule
A judgment that resolves only some of your claims, leaving the rest of the case still pending, is what the Rules of Court call a several, separate or partial judgment. A several, separate or partial judgment may be executed under the same terms and conditions as execution of a judgment or final order pending appeal. In other words, there is no separate, easier standard for partial judgments; the same rules that govern executing a whole judgment before the appeal period runs out apply here too.
You have to move for it, and show good reasons
Execution ahead of a final resolution is never automatic. Discretionary execution may only issue upon good reasons to be stated in a special order after due hearing. The prevailing party has to file a motion, with notice to the other side, in the court where the case is pending, and that court has to actually find good reasons for allowing execution now rather than waiting, and put those reasons in a written order.
Where the motion has to be filed
Timing and forum matter. The motion for execution pending appeal, or for execution of your partial judgment, has to be filed in the trial court while it has jurisdiction over the case and is in possession of either the original record or the record on appeal. Once the trial court has lost jurisdiction over the case, typically once the rest of the case has moved up on appeal, the motion has to be brought before the appellate court instead.
What this means for the rest of your case
Executing a partial judgment does not resolve or accelerate the claims that are still pending. Those continue through the ordinary litigation process, and only the portion already decided moves toward enforcement, and only if the court is satisfied there are good reasons to let that happen now instead of after the case is completely over. Whether your particular claims qualify depends on the specific reasons you can show the court for not waiting. Good reasons commonly involve urgency, such as a losing party's apparent intent to dissipate assets before the rest of the case can be finished, but the court still weighs each request on its own facts.