I just took at look at the v1.5.1 vs v1.5.2 code, and I think the majority of compatibility problems between PHP 5.3 to PHP 5.6 can be fixed by copying over (merging any customizations you've made to these files, if any) from the v1.5.2-beta files into your 1.5.1 site:
The most critical components (I think, off the top of my head at the moment) are: the mysql-to-mysqli change, timezone change, and changes to the way the SESSION[shipping] stuff is done to accommodate stricter array handling by adding checks about SESSION[shipping][id].
There might be more, but these should cover the major issues, and buy you a bit of time.
/admin/includes/extra_configures/use_catalog_time_zone.php
/admin/includes/functions/database.php
/admin/includes/application_top.php
/admin/products_price_manager.php
/includes/classes/db/mysql/query_factory.php
/includes/classes/class.base.php
/includes/classes/order.php
/includes/classes/payment.php
/includes/classes/query_cache.php
/includes/classes/shipping.php
/includes/extra_configures/set_time_zone.php
/includes/functions/password_funcs.php
/includes/init_includes/init_file_db_names.php
/includes/modules/order_total/ot_shipping.php
/includes/modules/pages/checkout_confirmation/header_php.php
/includes/modules/pages/checkout_payment/header_php.php
/includes/modules/pages/checkout_shipping/header_php.php
/includes/modules/pages/checkout_shipping_address/header_php.php
/includes/modules/pages/download/header_php.php
/includes/modules/payment/linkpoint_api/class.linkpoint_api.php
/includes/modules/checkout_new_address.php
/includes/templates/template_default/templates/tpl_checkout_shipping_default.php
/includes/application_top.php
(maybe) /includes/modules/shipping_estimator.php if you're using it and getting errors from it.
plus the changes shown in these:
https://github.com/zencart/zencart/pull/282/files
https://github.com/zencart/zencart/pull/492/files
https://github.com/zencart/zencart/pull/494/files?w=1
And maybe a couple fixes in your payment/shipping modules, depending which ones you're using:
https://github.com/zencart/zencart/commit/cb83287ada35605e20e06b8fe614555b5a729460
Plus the more recently-posted CURL parameter fixes posted to mitigate against POODLE etc.
The above changes should not affect anything in your database, so there's no need to make any database changes for this "patch".
BUT after the site is once again working on the new PHP version one should do a complete and proper upgrade including database updates via zc_install. http://www.zen-cart.com/entry.php?3-How-do-I-rebuild-my-site-on-the-new-version-instead-of-upgrading
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