First congrats on caring about the landscape in your care and wanting to improve your rental space. The cheapest way to control weeds without herbicides is wet newspaper. Lay sections of newspaper several pages thick on top of weedy areas to stop flowering weeds from spreading seeds all about. Then each time you visit the grocery store put one item back and use the money you save to buy one bag of bark mulch to put on top of the  newspaper. Next show the landlord that you are improving the property and conserving water with the mulch and ask for a delivery of more mulch. Weeds need sunlight so covering them with bark, gravel or wood chips (sometimes you can get free wood chips from an arborist) will control the infestation. Hand weeding is the easy answer for instant gratification and something children can do with you. In the lawn use a dull kitchen knife to dig out weeds turning it into a completion or "the backyard Olympics" to get your kids involved. Perhaps the owner will allow lawn aeration and mowing higher to help the lawn crowd out the weeds. Weeding early and often is the way to get ahead of a weed infested property.  Work at it this summer and next spring get out there and pull and smother weeds as soon as you see new growth. The rewards next summer will be amazing. Keep growing - but not weeds,  Marianne Binetti